A Short History of Fractal Dimension

It is second nature to think of integer dimensions:  A line is one dimensional.  A plane is two dimensional. A volume is three dimensional.  A point has no dimensions.

It is harder to think in four dimensions and higher, but even here it is a simple extrapolation of lower dimensions.  Consider the basis vectors spanning a three-dimensional space consisting of the triples of numbers

Then a four dimensional hyperspace is just created by adding a new “tuple” to the list

and so on to 5 and 6 dimensions and on.  Child’s play!

But how do you think of fractional dimensions?  What is a fractional dimension?  For that matter, what is a dimension?  Even the integer dimensions began to unravel when George Cantor showed in 1877 that the line and the plane, which clearly had different “dimensionalities”, both had the same cardinality and could be put into a one-to-one correspondence.  From then onward the concept of dimension had to be rebuilt from the ground up, leading ultimately to fractals.

Here is a short history of fractal dimension, partially excerpted from my history of dynamics in Galileo Unbound (Oxford University Press, 2018) pg. 110 ff.  This blog page presents the history through a set of publications that successively altered how mathematicians thought about curves in spaces, beginning with Karl Weierstrass in 1872.

Karl Weierstrass (1872)

Karl Weierstrass (1815 – 1897) was studying convergence properties of infinite power series in 1872 when he began with a problem that Bernhard Riemann had given to his students some years earlier.  Riemann had asked whether the function

was continuous everywhere but not differentiable.  This simple question about a simple series was surprisingly hard to answer (it was not solved until Hardy provided the proof in 1916 [1]).  Therefore, Weierstrass conceived of a simpler infinite sum that was continuous everywhere and for which he could calculate left and right limits of derivatives at any point.  This function is

where b is a large odd integer and a is positive and less than one.  Weierstrass showed that the left and right derivatives failed to converge to the same value, no matter where he took his point.  In short, he had discovered a function that was continuous everywhere, but had a derivative nowhere [2].  This pathological function, called a “Monster” by Charles Hermite, is now called the Weierstrass function.

Beyond the strange properties that Weierstrass sought, the Weierstrass function would turn out to be a fractal curve (recognized much later by Besicovitch and Ursell in 1937 [3]) with a fractal (Hausdorff) dimension given by

although this was not proven until very recently [4].  An example of the function is shown in Fig. 1 for a = 0.5 and b = 5.  This specific curve has a fractal dimension D = 1.5693.  Notably, this is a number that is greater than 1 dimension (the topological dimension of the curve) but smaller than 2 dimensions (the embedding dimension of the curve).  The curve tends to fill more of the two dimensional plane than a straight line, so its intermediate fractal dimension has an intuitive feel about it.  The more “monstrous” the curve looks, the closer its fractal dimension approaches 2.

Fig. 1  Weierstrass’ “Monster” (1872) with a = 0.5, b = 5.  This continuous function is nowhere differentiable.  It is a fractal with fractal dimension D = 2 + ln(0.5)/ln(5) = 1.5693.

Georg Cantor (1883)

Partially inspired by Weierstrass’ discovery, George Cantor (1845 – 1918) published an example of an unusual ternary set in 1883 in “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (“Foundations of a General Theory of Aggregates”) [5].  The set generates a function (The Cantor Staircase) that has a derivative equal to zero almost everywhere, yet whose area integrates to unity.  It is a striking example of a function that is not equal to the integral of its derivative!  Cantor demonstrated that the size of his set is aleph0 , which is the cardinality of the real numbers.  But whereas the real numbers are uniformly distributed, Cantor’s set is “clumped”.  This clumpiness is an essential feature that distinguishes it from the one-dimensional number line, and it raised important questions about dimensionality. The fractal dimension of the ternary Cantor set is DH = ln(2)/ln(3) = 0.6309.

Fig. 2  The 1883 Cantor set (below) and the Cantor staircase (above, as the indefinite integral over the set).

Giuseppe Peano (1890)

In 1878, in a letter to his friend Richard Dedekind, Cantor showed that there was a one-to-one correspondence between the real numbers and the points in any n-dimensional space.  He was so surprised by his own result that he wrote to Dedekind “I see it, but I don’t believe it.”  The solid concepts of dimension and dimensionality were dissolving before his eyes.  What does it mean to trace the path of a trajectory in an n-dimensional space, if all the points in n dimensions were just numbers on a line?  What could such a trajectory look like?  A graphic example of a plane-filling path was constructed in 1890 by Peano [6], who was a peripatetic mathematician with interests that wandered broadly across the landscape of the mathematical problems of his day—usually ahead of his time.  Only two years after he had axiomatized linear vector spaces [7], Peano constructed a continuous curve that filled space. 

The construction of Peano’s curve proceeds by taking a square and dividing it into 9 equal sub squares.  Lines connect the centers of each of the sub squares.  Then each sub square is divided again into 9 sub squares whose centers are all connected by lines.  At this stage, the original pattern, repeated 9 times, is connected together by 8 links, forming a single curve.  This process is repeated infinitely many times, resulting in a curve that passes through every point of the original plane square.  In this way, a line is made to fill a plane.  Where Cantor had proven abstractly that the cardinality of the real numbers was the same as the points in n-dimensional space, Peano created a specific example.  This was followed quickly by another construction, invented by David Hilbert in 1891, that divided the square into four instead of nine, simplifying the construction, but also showing that such constructions were easily generated.

Fig. 3 Peano’s (1890) and Hilbert’s (1891) plane-filling curves.  When the iterations are taken to infinity, the curves approach every point of two-dimensional space arbitrarily closely, giving them a dimension DH = DE = 2, although their topological dimensions are DT = 1.

Helge von Koch (1904)

The space-filling curves of Peano and Hilbert have the extreme property that a one-dimensional curve approaches every point in a two-dimensional space.  This ability of a one-dimensional trajectory to fill space mirrored the ergodic hypothesis that Boltzmann relied upon as he developed statistical mechanics.  These examples by Peano, Hilbert and Boltzmann inspired searches for continuous curves whose dimensionality similarly exceeded one dimension, yet without filling space.  Weierstrass’ Monster was already one such curve, existing in some dimension greater than one but not filling the plane.  The construction of the Monster required infinite series of harmonic functions, and the resulting curve was single valued on its domain of real numbers. 

An alternative approach was proposed by Helge von Koch (1870—1924), a Swedish mathematician with an interest in number theory.  He suggested in 1904 that a set of straight line segments could be joined together, and then shrunk by a scale factor to act as new segments of the original pattern [8].  The construction of the Koch curve is shown in Fig. 4.  When the process is taken to its limit, it produces a curve, differentiable nowhere, which snakes through two dimensions.  When connected with other identical curves into a hexagon, the curve resembles a snowflake, and the construction is known as “Koch’s Snowflake”. 

The Koch curve begins in generation 1 with N0 = 4 elements.  These are shrunk by a factor of b = 1/3 to become the four elements of the next generation, and so on.  The number of elements varies with the observation scale according to the equation

where D is called the fractal dimension.  In the example of the Koch curve, the fractal dimension is

which is a number less than its embedding dimenion DE = 2.  The fractal is embedded in 2D but has a fractional dimension that is greater than it topological dimension DT = 1.

Fig. 4  Generation of a Koch curve (1904).  The fractal dimension is D = ln(4)/ln(3) = 1.26.  At each stage, four elements are reduced in size by a factor of 3.  The “length” of the curve approaches infinity as the features get smaller and smaller.  But the scaling of the length with size is determined uniquely by the fractal dimension.

Waclaw Sierpinski (1915)

Waclaw Sierpinski (1882 – 1969) was a Polish mathematician studying at the Jagellonian University in Krakow for his doctorate when he came across a theorem that every point in the plane can be defined by a single coordinate.  Intrigued by such an unintuitive result, he dived deep into Cantor’s set theory after he was appointed as a faculty member at the university in Lvov.  He began to construct curves that had more specific properties than the Peano or Hilbert curves, such as a curve that passes through every interior point of a unit square but that encloses an area that is only equal to 5/12 = 0.4167.  Sierpinski became interested in the topological properties of such sets.

Sierpinski considered how to define a curve that was embedded in DE = 2 but that was NOT constructed as a topological dimension DT = 1 curve as the curves of Peano, Hilbert, Koch (and even his own) had been.  To demonstrate this point, he described a construction that began with a topological dimension DT = 2 object, a planar triangle, from which the open set of its central inverted triangle is removed, leaving its boundary points.  The process is continued iteratively to all scales [9].  The resulting point set is shown in Fig. 5 and is called the Sierpinski gasket.  What is left after all the internal triangles are removed is a point set that can be made discontinuous by cutting it at a finite set of points.  This is shown in Fig. 5 by the red circles.  Each circle, no matter the size, cuts the set at three points, making the resulting set discontinuous.  Ten years later, Karl Menger would show that this property of discontinuous cuts determined the topological dimension of the Sierpinski gasket to be DT = 1.  The embedding dimension is of course DE = 2, and the fractal dimension of the Sierpinski gasket is

Fig. 5 The Sierpinski gasket.  The central triangle is removed (leaving its boundary) at each scale.  The pattern is self-similar with a fractal dimension DH = 1.5850.  Unintuitively, it has a topological dimension DT = 1.

Felix Hausdorff (1918)

The work by Cantor, Peano, von Koch and Sierpinski had created a crisis in geometry as mathematicians struggled to rescue concepts of dimensionality.  An important byproduct of that struggle was a much deeper understanding of concepts of space, especially in the hands of Felix Hausdorff. 

Felix Hausdorff (1868 – 1942) was born in Breslau, Prussia, and educated in Leipzig.  In his early years as a doctoral student, and as an assistant professor at Leipzig, he was a practicing mathematician by day and a philosopher and playwright by night, publishing under the pseudonym Paul Mongré.  He was at the University of Bonn working on set theory when the Greek mathematician Constatin Carathéodory published a paper in 1914 that showed how to construct a p-dimensional set in a q-dimensional space [9].  Haussdorff realized that he could apply similar ideas to the Cantor set.  He showed that the outer measure of the Cantor set would go discontinuously from zero to infinity as the fractional dimension increased smoothly.  The critical value where the measure changed its character became known as the Hausdorff dimension [11]

For the Cantor ternary set, the Hausdorff dimension is exactly DH = ln(2)/ln(3) = 0.6309.  This value for the dimension is less than the embedding dimension DE = 1 of the support (the real numbers on the interval [0, 1]), but it is also greater than DT = 0 which would hold for a countable number of points on the interval.  The work by Hausdorff became well known in the mathematics community who applied the idea to a broad range of point sets like Weierstrass’s monster and the Koch curve.

It is important to keep a perspective of what Hausdorff’s work meant during which period of time.  For instance, although the curves of Weierstrass, von Koch and Sierpinski were understood to present a challenge to concepts of dimension, it was only after Haussdorff that mathematicians began to think in terms of fractional dimensions and to calculate the fractional dimensions of these earlier point sets.  Despite the fact that Sierpinski created one of the most iconic fractals that we use as an example every day, he was unaware at the time that he was doing so.  His interest was topological—creating a curve for which any cut at any point would create disconnected subsets starting with objects (triangles) with topological dimension DT = 2.  In this way, talking about the early fractal objects tends to be anachronistic, using language to describe them that had not yet been invented at that time.

This perspective is also true for the ideas of topological dimension.  For instance, even Sierpinski was not fully tuned into the problems of defining topological dimension.  It turns out that what he created was a curve of topological dimension DT = 1, but that would only become clear later with the work of the Austrian mathematician Karl Menger.

Karl Menger (1926)

The day that Karl Menger (1902 – 1985) was born, his father, Carl Menger (1840 – 1941) lost his job.  Carl Menger was one of the founders of the famous Viennese school that established the marginalist view of economics.  However, Carl was not married to Karl’s mother, which was frowned upon by polite Vienna society, so he had to relinquish his professorship.  Despite his father’s reduction in status, Karl received an excellent education at a Viennese gymnasium (high school).  Among of his classmates were Wolfgang Pauli (Nobel Prize for Physics in 1945)  and Richard Kuhn (Nobel Prize for Chemistry in 1938).  When Karl began attending the University of Vienna he studied physics, but the mathematics professor Hans Hahn opened his eyes to the fascinating work on analysis that was transforming mathematics at that time, so Karl shifted his studies to mathematical analysis, specifically concerning conceptions of “curves”. 

Menger made important contributions to the history of fractal dimension as well as the history of topological dimension.  In his approach to defining the intrinsic topological dimension of a point set, he described the construction of a point set embedded in three dimensions that had zero volume, an infinite surface area, and a fractal dimension between 2 and 3.  The object is shown in Fig. 6 and is called a Menger “sponge” [12].  The Menger sponge is a fractal with a fractal dimension DH = ln(20)/ln(3) = 2.7268.  The face of the sponge is also known as the Sierpinski carpt.  The fractal dimension of the Sierpinski carpet is DH = ln(8)/ln(3) = 1.8928.

Fig. 6 Menger Sponge. Embedding dimension DE = 3. Fractal dimension DH = ln(20)/ln(3) = 2.7268. Topological dimension DT = 1: all one-dimensional metric spaces can be contained within the Menger sponge point set. Each face is a Sierpinski carpet with fractal dimension DH = ln(8)/ln(3) = 1.8928.

The striking feature of the Menger sponge is its topological dimension.  Menger created a new definition of topological dimension that partially solved the crises created by Cantor when he showed that every point on the unit square can be defined by a single coordinate.  This had put a one dimensional curve in one-to-one correspondence with a two-dimensional plane.  Yet the topology of a 2-dimensional object is clearly different than the topology of a line.  Menger found a simple definition that showed why 2D is different, topologically, than 3D, despite Cantor’s conundrum.  The answer came from the idea of making cuts on a point set and seeing if the cut created disconnected subsets. 

As a simple example, take a 1D line.  The removal of a single point creates two disconnected sub-lines.  The intersection of the cut with the line is 0-dimensional, and Menger showed that this defined the line as 1-dimensional.  Similarly, a line cuts the unit square into to parts.  The intersection of the cut with the plane is 1-dimensional, signifying that the dimension of the plane is 2-dimensional.  In other words, a (n-1) dimensional intersection of the boundary of a small neighborhood with the point set indicates that the point set has a dimension of n.  Generalizing this idea, looking at the Sierpinski gasket in Fig. 5, the boundary of a small circular region, if placed appropriately (as in the figure), intersects the Sierpinski gasket at three points of dimension zero.  Hence, the topological dimension of the Sierpinski gasket is one-dimensional.  Manger was likewise able to show that his sponge also had a topology that was one-dimensional, DT = 1, despite the embedding dimension of DE = 3.  In fact, all 1-dimensional metric spaces can be fit inside a Menger Sponge.

Benoit Mandelbrot (1967)

Benoit Mandelbrot (1924 – 2010) was born in Warsaw and his family emigrated to Paris in 1935.  He attended the Ecole Polytechnique where he studied under Gaston Julia (1893 – 1978) and Paul Levy (1886 – 1971).  Both Julia and Levy made significant contributions to the field of self-similar point sets and made a lasting impression on Mandelbrot.  He went to Cal Tech for a master’s degree in aeronautics and then a PhD in mathematical sciences from the University of Paris.  In 1958 Mandelbrot joined the research staff of the IBM Thomas J. Watson Research Center in Yorktown Heights, New York where he worked for over 35 years on topics of information theory and economics, always with a view of properties of self-similar sets and time series.

In 1967 Mandelbrot published one of his early important papers on the self-similar properties of the coastline of Britain.  He proposed that many natural features had statistical self similarity, which he applied to coastlines.  He published the work as “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension[13] in Science magazine , where he showed that the length of the coastline diverged with a Hausdorff dimension equal to D = 1.25.  Working at IBM, a world leader in computers, he had ready access to their power as well as their visualization capabilities.  Therefore, he was one of the first to begin exploring the graphical character of self-similar maps and point sets.

During one of his sabbaticals at Harvard University he began exploring the properties of Julia sets (named after his former teacher at the Ecole Polytechnique).  The Julia set is a self-similar point set that is easily visualized in the complex plane (two dimensions).  As Mandelbrot studied the convergence of divergence of infinite series defined by the Julia mapping, he discovered an infinitely nested pattern that was both beautiful and complex.  This has since become known as the Mandelbrot set.

Fig. 7 Mandelbrot set.

Later, in 1975, Mandelbrot coined the term fractal to describe these self-similar point sets, and he began to realize that these types of sets were ubiquitous in nature, ranging from the structure of trees and drainage basins, to the patterns of clouds and mountain landscapes.  He published his highly successful and influential book The Fractal Geometry of Nature in 1982, introducing fractals to the wider public and launching a generation of hobbyists interested in computer-generated fractals.  The rise of fractal geometry coincided with the rise of chaos theory that was aided by the same computing power.  For instance, important geometric structures of chaos theory, known as strange attractors, have fractal geometry. 

Appendix:  Box Counting

When confronted by a fractal of unknown structure, one of the simplest methods to find the fractal dimension is through box counting.  This method is shown in Fig. 8.  The fractal set is covered by a set of boxes of size b, and the number of boxes that contain at least one point of the fractal set are counted.  As the boxes are reduced in size, the number of covering boxes increases as 

To be numerically accurate, this method must be iterated over several orders of magnitude.  The number of boxes covering a fractal has this characteristic power law dependence, as shown in Fig. 8, and the fractal dimension is obtained as the slope.

Fig. 8  Calculation of the fractal dimension using box counting.  At each generation, the size of the grid is reduced by a factor of 3.  The number of boxes that contain some part of the fractal curve increases as  , where b is the scale


[1] Hardy, G. (1916). “Weierstrass’s non-differentiable function.” Transactions of the American Mathematical Society 17: 301-325.

[2] Weierstrass, K. (1872). “Uber continuirliche Functionen eines reellen Arguments, die fur keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen.” Communication ri I’Academie Royale des Sciences II: 71-74.

[3] Besicovitch, A. S. and H. D. Ursell (1937). “Sets of fractional dimensions: On dimensional numbers of some continuous curves.” J. London Math. Soc. 1(1): 18-25.

[4] Shen, W. (2018). “Hausdorff dimension of the graphs of the classical Weierstrass functions.” Mathematische Zeitschrift. 289(1–2): 223–266.

[5] Cantor, G. (1883). Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Leipzig, B. G. Teubner.

[6] Peano, G. (1890). “Sur une courbe qui remplit toute une aire plane.” Mathematische Annalen 36: 157-160.

[7] Peano, G. (1888). Calcolo geometrico secundo l’Ausdehnungslehre di H. Grassmann e precedutto dalle operazioni della logica deduttiva. Turin, Fratelli Bocca Editori.

[8] Von Koch, H. (1904). “Sur.une courbe continue sans tangente obtenue par une construction geometrique elementaire.” Arkiv for Mathematik, Astronomi och Fysich 1: 681-704.

[9] Sierpinski, W. (1915). “Sur une courbe dont tout point est un point de ramification.” Comptes Rendus Hebdomadaires des Seances de l’Academie des Sciences de Paris 160: 302-305.

[10] Carathéodory, C. (1914). “Über das lineare Mass von Punktmengen – eine Verallgemeinerung des Längenbegriffs.” Gött. Nachr. IV: 404–406.

[11] Hausdorff, F. (1919). “Dimension und ausseres Mass.” Mathematische Anna/en 79: 157-179.

[12] Menger, Karl (1926), “Allgemeine Räume und Cartesische Räume. I.”, Communications to the Amsterdam Academy of Sciences. English translation reprinted in Edgar, Gerald A., ed. (2004), Classics on fractals, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO

[13] B Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, 156 3775 (May 5, 1967): 636-638.

The Ups and Downs of the Compound Double Pendulum

A chief principle of chaos theory states that even simple systems can display complex dynamics.  All that is needed for chaos, roughly, is for a system to have at least three dynamical variables plus some nonlinearity. 

A classic example of chaos is the driven damped pendulum.  This is a mass at the end of a massless rod driven by a sinusoidal perturbation.  The three variables are the angle, the angular velocity and the phase of the sinusoidal drive.  The nonlinearity is provided by the cosine function in the potential energy which is anharmonic for large angles.  However, the driven damped pendulum is not an autonomous system, because the drive is an external time-dependent function.  To find an autonomous system—one that persists in complex motion without any external driving function—one needs only to add one more mass to a simple pendulum to create what is known as a compound pendulum, or a double pendulum.

Daniel Bernoulli and the Discovery of Normal Modes

After the invention of the calculus by Newton and Leibniz, the first wave of calculus practitioners (Leibniz, Jakob and Johann Bernoulli and von Tschirnhaus) focused on static problems, like the functional form of the catenary (the shape of a hanging chain), or on constrained problems, like the brachistochrone (the path of least time for a mass under gravity to move between two points) and the tautochrone (the path of equal time).

The next generation of calculus practitioners (Euler, Johann and Daniel Bernoulli, and  D’Alembert) focused on finding the equations of motion of dynamical systems.  One of the simplest of these, that yielded the earliest equations of motion as well as the first identification of coupled modes, was the double pendulum.  The double pendulum, in its simplest form, is a mass on a rigid massless rod attached to another mass on a massless rod.  For small-angle motion, this is a simple coupled oscillator.

Fig. 1 The double pendulum as seen by Daniel Bernoulli, Johann Bernoulli and D’Alembert. This two-mass system played a central role in the earliest historical development of dynamical equations of motion.

Daniel Bernoulli, the son of Johann I Bernoulli, was the first to study the double pendulum, publishing a paper on the topic in 1733 in the proceedings of the Academy in St. Petersburg just as he returned from Russia to take up a post permanently in his home town of Basel, Switzerland.  Because he was a physicist first and mathematician second, he performed experiments with masses on strings to attempt to understand the qualitative as well as quantitative behavior of the two-mass system.  He discovered that for small motions there was a symmetric behavior that had a low frequency of oscillation and an antisymmetric motion that had a higher frequency of oscillation.  Furthermore, he recognized that any general motion of the double pendulum was a combination of the fundamental symmetric and antisymmetric motions.  This work by Daniel Bernoulli represents the discovery of normal modes of coupled oscillators.  It is also the first statement of the combination of motions that he would use later (1753) to express for the first time the principle of superposition. 

Superposition is one of the guiding principles of linear physical systems.  It provides a means for the solution of differential equations.  It explains the existence of eigenmodes and their eigenfrequencies.  It is the basis of all interference phenomenon, whether classical like the Young’s double-slit experiment or quantum like Schrödinger’s cat.  Today, superposition has taken center stage in quantum information sciences and helps define the spooky (and useful) properties of quantum entanglement.  Therefore, normal modes, composition of motion, superposition of harmonics on a musical string—these all date back to Daniel Bernoulli in the twenty years between 1733 and 1753.  (Daniel Bernoulli is also the originator of the Bernoulli principle that explains why birds and airplanes fly.)

Johann Bernoulli and the Equations of Motion

Daniel Bernoulli’s father was Johann I Bernoulli.  Daniel had been tutored by Johann, along with his friend Leonhard Euler, when Daniel was young.  But as Daniel matured as a mathematician, he and his father began to compete against each other in international mathematics competitions (which were very common in the early eighteenth century).  When Daniel beat his father in a competition sponsored by the French Academy, Johann threw Daniel out of his house and their relationship remained strained for the remainder of their lives.

Johann had a history of taking ideas from Daniel and never citing the source. For instance, when Johann published his work on equations of motion for masses on strings in 1742, he built on the work of his son Daniel from 1733 but never once mentioned it. Daniel, of course, was not happy.

In a letter dated 20 October 1742 that Daniel wrote to Euler, he said, “The collected works of my father are being printed, and I have Just learned that he has inserted, without any mention of me, the dynamical problems I first discovered and solved (such as e. g. the descent of a sphere on a moving triangle; the linked pendulum, the center of spontaneous rotation, etc.).” And on 4 September 1743, when Daniel had finally seen his father’s works in print, he said, “The new mechanical problems are mostly mine, and my father saw my solutions before he solved the problems in his way …”. [2]

Daniel clearly has the priority for the discovery of the normal modes of the linked (i.e. double or compound) pendulum, but Johann often would “improve” on Daniel’s work despite giving no credit for the initial work. As a mathematician, Johann had a more rigorous approach and could delve a little deeper into the math. For this reason, it was Johann in 1742 who came closest to writing down differential equations of motion for multi-mass systems, but falling just short. It was D’Alembert only one year later who first wrote down the differential equations of motion for systems of masses and extended it to the loaded string for which he was the first to derive the wave equation. The D’Alembertian operator is today named after him.

Double Pendulum Dynamics

The general dynamics of the double pendulum are best obtained from Lagrange’s equations of motion. However, setting up the Lagrangian takes careful thought, because the kinetic energy of the second mass depends on its absolute speed which is dependent on the motion of the first mass from which it is suspended. The velocity of the second mass is obtained through vector addition of velocities.

Fig. 2. The dynamics of the double pendulum.

The potential energy of the system is

so that the Lagrangian is

The partial derivatives are

and the time derivatives of the last two expressions are

Therefore, the equations of motion are

To get a sense of how this system behaves, we can make a small-angle approximation to linearize the equations to find the lowest-order normal modes.  In the small-angle approximation, the equations of motion become

where the determinant is

This quartic equation is quadratic in w2 and the quadratic solution is

This solution is still a little opaque, so taking the special case: R = R1 = R2 and M = M1 = M2 it becomes

There are two normal modes.  The low-frequency mode is symmetric as both masses swing (mostly) together, while the higher frequency mode is antisymmetric with the two masses oscillating against each other.  These are the motions that Daniel Bernoulli discovered in 1733.

It is interesting to note that if the string were rigid, so that the two angles were the same, then the lowest frequency would be 3/5 which is within 2% of the above answer but is certainly not equal.  This tells us that there is a slightly different angular deflection for the second mass relative to the first.

Chaos in the Double Pendulum

The full expression for the nonlinear coupled dynamics is expressed in terms of four variables (q1, q2, w1, w2).  The dynamical equations are

These can be put into the normal form for a four-dimensional flow as

The numerical solution of these equations produce a complex interplay between the angle of the first mass and the angle of the second mass. Examples of trajectory projections in configuration space are shown in Fig. 3 for E = 1. The horizontal is the first angle, and the vertical is the angle of the second mass.

Fig. 3 Trajectory projections onto configuration space. The horizontal axis is the first mass angle, and the vertical axis is the second mass angle. All of these are periodic or nearly periodic orbits except for the one on the lower left. E = 1.

The dynamics in state space are four dimensional which are difficult to visualize directly. Using the technique of the Poincaré first-return map, the four-dimensional trajectories can be viewed as a two-dimensional plot where the trajectories pierce the Poincaré plane. Poincare sections are shown in Fig. 4.

Fig. Poincare sections of the double pendulum in state space for increasing kinetic energy. Initial conditions are vertical in all. The horizontal axis is the angle of the second mass, and the vertical axis is the angular velocity of the second mass.

Python Code: DoublePendulum.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
Created on Oct 16 06:03:32 2020
"Introduction to Modern Dynamics" 2nd Edition (Oxford, 2019)
@author: nolte

import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
import time


E = 1.       # Try 0.8 to 1.5

def flow_deriv(x_y_z_w,tspan):
    x, y, z, w = x_y_z_w

    A = w**2*np.sin(y-x);
    B = -2*np.sin(x);
    C = z**2*np.sin(y-x)*np.cos(y-x);
    D = np.sin(y)*np.cos(y-x);
    EE = 2 - (np.cos(y-x))**2;
    FF = w**2*np.sin(y-x)*np.cos(y-x);
    G = -2*np.sin(x)*np.cos(y-x);
    H = 2*z**2*np.sin(y-x);
    I = 2*np.sin(y);
    JJ = (np.cos(y-x))**2 - 2;

    a = z
    b = w
    c = (A+B+C+D)/EE
    d = (FF+G+H+I)/JJ

repnum = 75

for reploop  in range(repnum):
    px1 = 2*(np.random.random((1))-0.499)*np.sqrt(E);
    py1 = -px1 + np.sqrt(2*E - px1**2);            

    xp1 = 0   # Try 0.1
    yp1 = 0   # Try -0.2
    x_y_z_w0 = [xp1, yp1, px1, py1]
    tspan = np.linspace(1,1000,10000)
    x_t = integrate.odeint(flow_deriv, x_y_z_w0, tspan)
    siztmp = np.shape(x_t)
    siz = siztmp[0]

    if reploop % 50 == 0:
        lines = plt.plot(x_t[:,0],x_t[:,1])
        plt.setp(lines, linewidth=0.5)

    y1 = np.mod(x_t[:,0]+np.pi,2*np.pi) - np.pi
    y2 = np.mod(x_t[:,1]+np.pi,2*np.pi) - np.pi
    y3 = np.mod(x_t[:,2]+np.pi,2*np.pi) - np.pi
    y4 = np.mod(x_t[:,3]+np.pi,2*np.pi) - np.pi
    py = np.zeros(shape=(10*repnum,))
    yvar = np.zeros(shape=(10*repnum,))
    cnt = -1
    last = y1[1]
    for loop in range(2,siz):
        if (last < 0)and(y1[loop] > 0):
            cnt = cnt+1
            del1 = -y1[loop-1]/(y1[loop] - y1[loop-1])
            py[cnt] = y4[loop-1] + del1*(y4[loop]-y4[loop-1])
            yvar[cnt] = y2[loop-1] + del1*(y2[loop]-y2[loop-1])
            last = y1[loop]
            last = y1[loop]
    lines = plt.plot(yvar,py,'o',ms=1)

You can change the energy E on line 16 and also the initial conditions xp1 and yp1 on lines 48 and 49. The energy E is the initial kinetic energy imparted to the two masses. For a given initial condition, what happens to the periodic orbits as the energy E increases?


[1] Daniel Bernoulli, Theoremata de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae,” Academiae Scientiarum Imperialis Petropolitanae, 6, 1732/1733

[2] Truesdell B. The rational mechanics of flexible or elastic bodies, 1638-1788. (Turici: O. Fussli, 1960). (This rare and artistically produced volume, that is almost impossible to find today in any library, is one of the greatest books written about the early history of dynamics.)

The Bountiful Bernoullis of Basel

The task of figuring out who’s who in the Bernoulli family is a hard nut to crack.  The Bernoulli name populates a dozen different theorems or physical principles in the history of science and mathematics, but each one was contributed by any of four or five different Bernoullis of different generations—brothers, uncles, nephews and cousins.  What makes the task even more difficult is that any given Bernoulli might be called by several different aliases, while many of them shared the same name across generations.  To make things worse, they often worked and published on each other’s problems.

To attribute a theorem to a Bernoulli is not too different from attributing something to the famous mathematical consortium called Nicholas Bourbaki.  It’s more like a team rather than an individual.  But in the case of Bourbaki, the goal was selfless anonymity, while in the case of the Bernoullis it was sometimes the opposite—bald-faced competition and one-up-manship coupled with jealousy and resentment. Fortunately, the competition tended to breed more output than less, and the world benefited from the family feud.

The Bernoulli Family Tree

The Bernoullis are intimately linked with the beautiful city of Basel, Switzerland, situated on the Rhine River where it leaves Switzerland and forms the border between France and Germany . The family moved there from the Netherlands in the 1600’s to escape the Spanish occupation.

Basel Switzerland

The first Bernoulli born in Basel was Nikolaus Bernoulli (1623 – 1708), and he had four sons: Jakob I, Nikolaus, Johann I and Hieronymous I. The “I”s in this list refer to the fact, or the problem, that many of the immediate descendants took their father’s or uncle’s name. The long-lived family heritage in the roles of mathematician and scientist began with these four brothers. Jakob Bernoulli (1654 – 1705) was the eldest, followed by Nikolaus Bernoulli (1662 – 1717), Johann Bernoulli (1667 – 1748) and then Hieronymous (1669 – 1760). In this first generation of Bernoullis, the great mathematicians were Jakob and Johann. More mathematical equations today are named after Jakob, but Johann stands out because of the longevity of his contributions, the volume and impact of his correspondence, the fame of his students, and the number of offspring who also took up mathematics. Johann was also the worst when it came to jealousy and spitefulness—against his brother Jakob, whom he envied, and specifically against his son Daniel, whom he feared would eclipse him.

Jakob Bernoulli (aka James or Jacques or Jacob)

Jakob Bernoulli (1654 – 1705) was the eldest of the first generation of brothers and also the first to establish himself as a university professor. He held the chair of mathematics at the university in Basel. While his interests ranged broadly, he is known for his correspondences with Leibniz as he and his brother Johann were among the first mathematicians to apply Lebiniz’ calculus to solving specific problems. The Bernoulli differential equation is named after him. It was one of the first general differential equations to be solved after the invention of the calculus. The Bernoulli inequality is one of the earliest attempts to find the Taylor expansion of exponentiation, which is also related to Bernoulli numbers, Bernoulli polynomials and the Bernoulli triangle. A special type of curve that looks like an ellipse with a twist in the middle is the lemniscate of Bernoulli.

Perhaps Jakob’s most famous work was his Ars Conjectandi (1713) on probability theory. Many mathematical theorems of probability named after a Bernoulli refer to this work, such as Bernoulli distribution, Bernoulli’s golden theorem (the law of large numbers), Bernoulli process and Bernoulli trial.

Fig. Bernoulli numbers in Jakob’s Ars Conjectandi (1713)

Johann Bernoulli (aka Jean or John)

Jakob was 13 years older than his brother Johann Bernoulli (1667 – 1748), and Jakob tutored Johann in mathematics who showed great promise. Unfortunately, Johann had that awkward combination of high self esteem with low self confidence, and he increasingly sought to show that he was better than his older brother. As both brothers began corresponding with Leibniz on the new calculus, they also began to compete with one another. Driven by his insecurity, Johann also began to steal ideas from his older brother and claim them for himself.

A classic example of this is the famous brachistrochrone problem that was posed by Johann in the Acta Eruditorum in 1696. Johann at this time was a professor of mathematics at Gronigen in the Netherlands. He challenged the mathematical world to find the path of least time for a mass to travel under gravity between two points. He had already found one solution himself and thought that no-one else would succeed. Yet when he heard his brother Jakob was responding to the challenge, he spied out his result and then claimed it as his own. Within the year and a half there were 4 additional solutions—all correct—using different approaches.  One of the most famous responses was by Newton (who as usual did not give up his method) but who is reported to have solved the problem in a day.  Others who contributed solutions were Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l’Hôpital in addition to Jakob.

The participation of de l’Hôpital in the challenge was a particular thorn in Johann’s side, because de l’Hôpital had years earlier paid Johann to tutor him in Leibniz’ new calculus at a time when l’Hôpital knew nothing of the topic. What is today known as l’Hôpital’s theorem on ratios of limits in fact was taught to l’Hôpital by Johann. Johann never forgave l’Hôpital for publicizing the result—but l’Hôpital had the discipline to write a textbook while Johann did not. To be fair, l’Hôpital did give Johann credit in the opening of his book, but that was not enough for Johann who continued to carry his resentment.

When Jakob died of tuberculosis in 1705, Johann campaigned to replace him in his position as professor of mathematics and succeeded. In that chair, Johann had many famous students (Euler foremost among them, but also Maupertuis and Clairaut). Part of Johann’s enduring fame stems from his many associations and extensive correspondences with many of the top mathematicians of the day. For instance, he had a regular correspondence with the mathematician Varignon, and it was in one of these letters that Johann proposed the principle of virtual velocities which became a key axiom for Joseph Lagrange’s later epic work on the foundations of mechanics (see Chapter 4 in Galileo Unbound).

Johann remained in his chair of mathematics at Basel for almost 40 years. This longevity, and the fame of his name, guaranteed that he taught some of the most talented mathematicians of the age, including his most famous student Leonhard Euler, who is held by some as one of the four greatest mathematicians of all time (the others were Archimedes, Newton and Gauss) [1].

Nikolaus I Bernoulli

Nikolaus I Bernoulli (1687 – 1759, son of Nikolaus) was the cousin of Daniel and nephew to both Jacob and Johann. He was a well-known mathematician in his time (he briefly held Galileo’s chair in Padua), though few specific discoveries are attributed to him directly. He is perhaps most famous today for posing the “St. Petersburg Paradox” of economic game theory. Ironically, he posed this paradox while his cousin Nikolaus II Bernoulli (brother of Daniel Bernoulli) was actually in St. Petersburg with Daniel.

The St. Petersburg paradox is a simple game of chance played with a fair coin where a player must buy in at a certain price in order to place $2 in a pot that doubles each time the coin lands heads, and pays out the pot at the first tail. The average pay-out of this game has infinite expectation, so it seems that anyone should want to buy in at any cost. But most people would be unlikely to buy in even for a modest $25. Why? And is this perception correct? The answer was only partially provided by Nikolaus. The definitive answer was given by his cousin Daniel Bernoulli.

Daniel Bernoulli

Daniel Bernoulli (1700 – 1782, son of Johann I) is my favorite Bernoulli. While most of the other Bernoullis were more mathematicians than scientists, Daniel Bernoulli was more physicist than mathematician. When we speak of “Bernoulli’s principle” today, the fundamental force that allows birds and airplanes to fly, we are referring to his work on hydrodynamics. He was one of the earliest originators of economic dynamics through his invention of the utility function and diminishing returns, and he was the first to clearly state the principle of superposition, which lies at the heart today of the physics of waves and quantum technology.

Daniel Bernoulli

While in St. Petersburg, Daniel conceived of the solution to the St. Petersburg paradox (he is the one who actually named it). To explain why few people would pay high stakes to play the game, he devised a “utility function” that had “diminishing marginal utility” in which the willingness to play depended on ones wealth. Obviously a wealthy person would be willing to pay more than a poor person. Daniel stated

The determination of the value of an item must not be based on the price, but rather on the utility it yields…. There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount.

He created a log utility function that allowed one to calculate the highest stakes a person should be willing to take based on their wealth. Indeed, a millionaire may only wish to pay $20 per game to play, in part because the average payout over a few thousand games is only about $5 per game. It is only in the limit of an infinite number of games (and an infinite bank account by the casino) that the average payout diverges.

Daniel Bernoulli Hydrodynamica (1638)

Johann II Bernoulli

Daniel’s brother Johann II (1710 – 1790) published in 1736 one of the most important texts on the theory of light during the time between Newton and Euler. Although the work looks woefully anachronistic today, it provided one of the first serious attempts at understanding the forces acting on light rays and describing them mathematically [5]. Euler based his new theory of light, published in 1746, on much of the work laid down by Johann II. Euler came very close to proposing a wave-like theory of light, complete with a connection between frequency of wave pulses and colors, that would have preempted Thomas Young by more than 50 years. Euler, Daniel and Johann II as well as Nicholas II were all contemporaries as students of Johann I in Basel.

More Relations

Over the years, there were many more Bernoullis who followed in the family tradition. Some of these include:

Johann II Bernoulli (1710–1790; also known as Jean), son of Johann, mathematician and physicist

Johann III Bernoulli (1744–1807; also known as Jean), son of Johann II, astronomer, geographer and mathematician

Jacob II Bernoulli (1759–1789; also known as Jacques), son of Johann II, physicist and mathematician

Johann Jakob Bernoulli (1831–1913), art historian and archaeologist; noted for his Römische Ikonographie (1882 onwards) on Roman Imperial portraits

Ludwig Bernoully (1873 – 1928), German architect in Frankfurt

Hans Bernoulli (1876–1959), architect and designer of the Bernoullihäuser in Zurich and Grenchen SO

Elisabeth Bernoulli (1873-1935), suffragette and campaigner against alcoholism.

Notable marriages to the Bernoulli family include the Curies (Pierre Curie was a direct descendant to Johann I) as well as the German author Hermann Hesse (married to a direct descendant of Johann I).


[1] Calinger, Ronald S.. Leonhard Euler : Mathematical Genius in the Enlightenment, Princeton University Press (2015).

[2] Euler L and Truesdell C. Leonhardi Euleri Opera Omnia. Series secunda: Opera mechanica et astronomica XI/2. The rational mechanics of flexible or elastic bodies 1638-1788. (Zürich: Orell Füssli, 1960).

[3] D Speiser, Daniel Bernoulli (1700-1782), Helvetica Physica Acta 55 (1982), 504-523.

[4] Leibniz GW. Briefwechsel zwischen Leibniz, Jacob Bernoulli, Johann Bernoulli und Nicolaus Bernoulli. (Hildesheim: Olms, 1971).

[5] Hakfoort C. Optics in the age of Euler : conceptions of the nature of light, 1700-1795. (Cambridge: Cambridge University Press, 1995).

Henri Poincaré and his Homoclinic Tangle

Will the next extinction-scale asteroid strike the Earth in our lifetime? 

This existential question—the question of our continued existence on this planet—is rhetorical, because there are far too many bodies in our solar system to accurately calculate all trajectories of all asteroids. 

The solar system is what is known as an N-body problem.  And even the N is not well determined.  The asteroid belt alone has over a million extinction-sized asteroids, and there are tens of millions of smaller ones that could still do major damage to life on Earth if they hit.  To have a hope of calculating even one asteroid trajectory do we ignore planetary masses that are too small?  What is too small?  What if we only consider the Sun, the Earth and Jupiter?  This is what Euler did in 1760, and he still had to make more assumptions.

Stability of the Solar System

Once Newton published his Principia, there was a pressing need to calculate the orbit of the Moon (see my blog post on the three-body problem).  This was important for navigation, because if the daily position of the moon could be known with sufficient accuracy, then ships would have a means to determine their longitude at sea.  However, the Moon, Earth and Sun are already a three-body problem, which still ignores the effects of Mars and Jupiter on the Moon’s orbit, not to mention the problem that the Earth is not a perfect sphere.  Therefore, to have any hope of success, toy systems that were stripped of all their obfuscating detail were needed.

Euler investigated simplified versions of the three-body problem around 1760, treating a body attracted to two fixed centers of gravity moving in the plane, and he solved it using elliptic integrals. When the two fixed centers are viewed in a coordinate frame that is rotating with the Sun-Earth system, it can come close to capturing many of the important details of the system. In 1762 Euler tried another approach, called the restricted three-body problem, where he considered a massless Moon attracted to a massive Earth orbiting a massive Sun, again all in the plane. Euler could not find general solutions to this problem, but he did stumble on an interesting special case when the three bodies remain collinear throughout their motions in a rotating reference frame.

It was not the danger of asteroids that was the main topic of interest in those days, but the question whether the Earth itself is in a stable orbit and is safe from being ejected from the Solar system.  Despite steadily improving methods for calculating astronomical trajectories through the nineteenth century, this question of stability remained open.

Poincaré and the King Oscar Prize of 1889

Some years ago I wrote an article for Physics Today called “The Tangled Tale of Phase Space” that tracks the historical development of phase space. One of the chief players in that story was Henri Poincaré (1854 – 1912). Henri Poincare was the Einstein before Einstein. He was a minor celebrity and was considered to be the greatest genius of his era. The event in his early career that helped launch him to stardom was a mathematics prize announced in 1887 to honor the birthday of King Oscar II of Sweden. The challenge problem was as simple as it was profound: Prove rigorously whether the solar system is stable.

This was the old N-body problem that had so far resisted solution, but there was a sense at that time that recent mathematical advances might make the proof possible. There was even a rumor that Dirichlet had outlined such a proof, but no trace of the outline could be found in his papers after his death in 1859.

The prize competition was announced in Acta Mathematica, written by the Swedish mathematician Gösta Mittag-Leffler. It stated:

Given a system of arbitrarily many mass points that attract each according to Newton’s law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.

The timing of the prize was perfect for Poincaré who was in his early thirties and just beginning to make his mark on mathematics. He was working on the theory of dynamical systems and was developing a new viewpoint that went beyond integrating single trajectories by focusing more broadly on whole classes of solutions. The question of the stability of the solar system seemed like a good problem to use to sharpen his mathematical tools. The general problem was still too difficult, so he began with Euler’s restricted three-body problem. He made steady progress, and along the way he invented an array of new techniques for studying the general properties of dynamical systems. One of these was the Poincaré section. Another was his set of integral invariants, one of which is recognized as the conservation of volume in phase space, also known as Liouville’s theorem, although it was Ludwig Boltzmann who first derived this result (see my Physics Today article). Eventually, he believed he had proven that the restricted three-body problem was stable.

By the time Poincaré had finished is prize submission, he had invented a new field of mathematical analysis, and the judges of the prize submission recognized it. Poincaré was named the winner, and his submission was prepared for publication in the Acta. However, Mittag-Leffler was a little concerned by a technical objection that had been raised, so he forwarded the comment to Poincaré for him to look at. At first, Poincaré thought the objection could easily be overcome, but as he worked on it and delved deeper, he had a sudden attack of panic. Trajectories near a saddle point did not converge. His proof of stability was wrong!

He alerted Mittag-Leffler to stop the presses, but it was too late. The first printing had been completed and review copies had already been sent to the judges. Mittag-Leffler immediately wrote to them asking for their return while Poincaré worked nonstop to produce a corrected copy. When he had completed his reanalysis, he had discovered a divergent feature of the solution to the dynamical problem near saddle points that his recognized today as the discovery of chaos. Poincaré paid for the reprinting of his paper out of his own pocket and (almost) all of the original printing was destroyed. This embarrassing moment in the life of a great mathematician was virtually forgotten until it was brought to light by the historian Barrow-Green in 1994 [1].

Poincaré is still a popular icon in France. Here is the Poincaré cafe in Paris.
A crater on the Moon is named after Poincaré.

Chaos in the Poincaré Return Map

Despite the fact that his conclusions on the stability of the 3-body problem flipped, Poincaré’s new tools for analyzing dynamical systems earned him the prize. He did not stop at his modified prize submission but continued working on systematizing his methods, publishing New Methods in Celestial Mechanics in several volumes through the 1890’s. It was here that he fully explored what happens when a trajectory approaches a saddle point of dynamical equilibrium.

The third volume of a three-book series that grew from Poincaré’s award-winning paper

To visualize a periodic trajectory, Poincaré invented a mathematical tool called a “first-return map”, also known as a Poincaré section. It was a way of taking a higher dimensional continuous trajectory and turning it into a simple iterated discrete map. Therefore, one did not need to solve continuous differential equations, it was enough to just iterate the map. In this way, complicated periodic, or nearly periodic, behavior could be explored numerically. However, even armed with this weapon, Poincaré found that iterated maps became unstable as a trajectory that originated from a saddle point approached another equivalent saddle point. Because the dynamics are periodic, the outgoing and incoming trajectories are opposite ends of the same trajectory, repeated with 2-pi periodicity. Therefore, the saddle point is also called a homoclinic point, meaning that trajectories in the discrete map intersect with themselves. (If two different trajectories in the map intersect, that is called a heteroclinic point.) When Poincaré calculated the iterations around the homoclinic point, he discovered a wild and complicated pattern in which a trajectory intersected itself many times. Poincaré wrote:

[I]f one seeks to visualize the pattern formed by these two curves and their infinite number of intersections … these intersections form a kind of lattice work, a weave, a chain-link network of infinitely fine mesh; each of the two curves can never cross itself, but it must fold back on itself in a very complicated way so as to recross all the chain-links an infinite number of times .… One will be struck by the complexity of this figure, which I am not even attempting to draw. Nothing can give us a better idea of the intricacy of the three-body problem, and of all the problems of dynamics in general…

Poincaré’s first view of chaos.

This was the discovery of chaos! Today we call this “lattice work” the “homoclinic tangle”. He could not draw it with the tools of his day … but we can!

Chirikov’s Standard Map

The restricted 3-body problem is a bit more complicated than is needed to illustrate Poincaré’s homoclinic tangle. A much simpler model is a discrete map called Chirikov’s Map or the Standard Map. It describes the Poincaré section of a periodically kicked oscillator that rotates or oscillates in the angular direction with an angular momentm J. The map has the simple form

in which the angular momentum in updated first, and then the angle variable is updated with the new angular momentum. When plotted on the (θ,J) plane, the standard map produces a beautiful kaleidograph of intertwined trajectories piercing the Poincaré plane, as shown in the figure below. The small points or dots are successive intersections of the higher-dimensional trajectory intersecting a plane. It is possible to trace successive points by starting very close to a saddle point (on the left) and connecting successive iterates with lines. These lines merge into the black trace in the figure that emerges along the unstable manifold of the saddle point on the left and approaches the saddle point on the right generally along the stable manifold.

Fig. Standard map for K = 0.97 at the transition to full chaos. The dark line is the trajectory of the unstable manifold emerging from the saddle point at (p,0). Note the wild oscillations as it approaches the saddle point at (3pi,0).

However, as the successive iterates approach the new saddle (which is really just the old saddle point because of periodicity) it crosses the stable manifold again and again, in ever wilder swings that diverge as it approaches the saddle point. This is just one trace. By calculating traces along all four stable and unstable manifolds and carrying them through to the saddle, a lattice work, or homoclinic tangle emerges.

Two of those traces originate from the stable manifolds, so to calculate their contributions to the homoclinic tangle, one must run these traces backwards in time using the inverse Chirikov map. This is

The four traces all intertwine at the saddle point in the figure below with a zoom in on the tangle in the next figure. This is the lattice work that Poincaré glimpsed in 1889 as he worked feverishly to correct the manuscript that won him the prize that established him as one of the preeminent mathematicians of Europe.

Fig. The homoclinic tangle caused by the folding of phase space trajectories as stable and unstable manifolds criss-cross in the Poincare map at the saddle point. This was the figure that Poincaré could not attempt to draw because of its complexity.
Fig. A zoom-in of the homoclinic tangle at the saddle point as the stable and unstable manifolds create a lattice of intersections. This is the fundamental origin of chaos and the sensitivity to initial conditions (SIC) that make forecasting almost impossible in chaotic systems.

Python Code: StandmapHom.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
Created on Sun Aug  2  2020
"Introduction to Modern Dynamics" 2nd Edition (Oxford, 2019)
@author: nolte

import numpy as np
from matplotlib import pyplot as plt
from numpy import linalg as LA


eps = 0.97



for eloop in range(0,100):

    rlast = 2*np.pi*(0.5-np.random.random())
    thlast = 4*np.pi*np.random.random()
    rplot = np.zeros(shape=(200,))
    thetaplot = np.zeros(shape=(200,))
    for loop in range(0,200):
        rnew = rlast + eps*np.sin(thlast)
        thnew = np.mod(thlast+rnew,4*np.pi)
        thetaplot[loop] = np.mod(thnew-np.pi,4*np.pi)     
        rtemp = np.mod(rnew + np.pi,2*np.pi)
        rplot[loop] = rtemp - np.pi
        rlast = rnew
        thlast = thnew

K = eps
eps0 = 5e-7

J = [[1,1+K],[1,1]]
w, v = LA.eig(J)

My = w[0]
Vu = v[:,0]     # unstable manifold
Vs = v[:,1]     # stable manifold

# Plot the unstable manifold
Hr = np.zeros(shape=(100,150))
Ht = np.zeros(shape=(100,150))
for eloop in range(0,100):
    eps = eps0*eloop

    roldu1 = eps*Vu[0]
    thetoldu1 = eps*Vu[1]
    Nloop = np.ceil(-6*np.log(eps0)/np.log(eloop+2))
    flag = 1
    cnt = 0
    while flag==1 and cnt < Nloop:
        ru1 = roldu1 + K*np.sin(thetoldu1)
        thetau1 = thetoldu1 + ru1
        roldu1 = ru1
        thetoldu1 = thetau1
        if thetau1 > 4*np.pi:
            flag = 0
        Hr[eloop,cnt] = roldu1
        Ht[eloop,cnt] = thetoldu1 + 3*np.pi
        cnt = cnt+1
x = Ht[0:99,12] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[0:99,12]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[5:39,15] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[5:39,15]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[12:69,16] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[12:69,16]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[15:89,17] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[15:89,17]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[30:99,18] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[30:99,18]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

# Plot the stable manifold
del Hr, Ht
Hr = np.zeros(shape=(100,150))
Ht = np.zeros(shape=(100,150))
#eps0 = 0.03
for eloop in range(0,100):
    eps = eps0*eloop

    roldu1 = eps*Vs[0]
    thetoldu1 = eps*Vs[1]
    Nloop = np.ceil(-6*np.log(eps0)/np.log(eloop+2))
    flag = 1
    cnt = 0
    while flag==1 and cnt < Nloop:
        thetau1 = thetoldu1 - roldu1
        ru1 = roldu1 - K*np.sin(thetau1)

        roldu1 = ru1
        thetoldu1 = thetau1
        if thetau1 > 4*np.pi:
            flag = 0
        Hr[eloop,cnt] = roldu1
        Ht[eloop,cnt] = thetoldu1
        cnt = cnt+1
x = Ht[0:79,12] + np.pi
x2 = 6*np.pi - x
y = Hr[0:79,12]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[4:39,15] + np.pi
x2 = 6*np.pi - x
y = Hr[4:39,15]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[12:69,16] + np.pi
x2 =  6*np.pi - x
y = Hr[12:69,16]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[15:89,17] + np.pi
x2 =  6*np.pi - x
y = Hr[15:89,17]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[30:99,18] + np.pi
x2 =  6*np.pi - x
y = Hr[30:99,18]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)


[1] D. D. Nolte, “The tangled tale of phase space,” Physics Today, vol. 63, no. 4, pp. 33-38, Apr (2010)

[2] M. C. Gutzwiller, “Moon-Earth-Sun: The oldest three-body problem,” Reviews of Modern Physics, vol. 70, no. 2, pp. 589-639, Apr (1998)

[3] Barrow-Green J. Oscar II’s Prize Competition and the Error in Poindare’s Memoir on the Three Body Problem. Arch Hist Exact Sci 48: 107-131, 1994.

[4] Barrow-Green J. Poincaré and the three body problem. London Mathematical Society, 1997.

[5] https://the-moon.us/wiki/Poincar%C3%A9

[6] Poincaré H and Goroff DL. New methods of celestial mechanics … Edited and introduced by Daniel L. Goroff. New York: American Institute of Physics, 1993.

Brook Taylor’s Infinite Series

When Leibniz claimed in 1704, in a published article in Acta Eruditorum, to have invented the differential calculus in 1684 prior to anyone else, the British mathematicians rushed to Newton’s defense. They knew Newton had developed his fluxions as early as 1666 and certainly no later than 1676. Thus ensued one of the most bitter and partisan priority disputes in the history of math and science that pitted the continental Leibnizians against the insular Newtonians. Although a (partisan) committee of the Royal Society investigated the case and found in favor of Newton, the affair had the effect of insulating British mathematics from Continental mathematics, creating an intellectual desert as the forefront of mathematical analysis shifted to France. Only when George Green filled his empty hours with the latest advances in French analysis, as he tended his father’s grist mill, did British mathematics wake up. Green self-published his epic work in 1828 that introduced what is today called Green’s Theorem.

Yet the period from 1700 to 1828 was not a complete void for British mathematics. A few points of light shone out in the darkness, Thomas Simpson, Collin Maclaurin, Abraham de Moivre, and Brook Taylor (1685 – 1731) who came from an English family that had been elevated to minor nobility by an act of Cromwell during the English Civil War.

Growing up in Bifrons House


View of Bifrons House from sometime in the late-1600’s showing the Jacobean mansion and the extensive south gardens.

When Brook Taylor was ten years old, his father bought Bifrons House [1], one of the great English country houses, located in the county of Kent just a mile south of Canterbury.  English country houses were major cultural centers and sources of employment for 300 years from the seventeenth century through the early 20th century. While usually being the country homes of nobility of all levels, from Barons to Dukes, sometimes they were owned by wealthy families or by representatives in Parliament, which was the case for the Taylors. Bifrons House had been built around 1610 in the Jacobean architectural style that was popular during the reign of James I.  The house had a stately front façade, with cupola-topped square towers, gable ends to the roof, porches of a renaissance form, and extensive manicured gardens on the south side.  Bifrons House remained the seat of the Taylor family until 1824 when they moved to a larger house nearby and let Bifrons first to a Marquess and then in 1828 to Lady Byron (ex-wife of Lord Byron) and her daughter Ada Lovelace (the mathematician famous for her contributions to early computer science). The Taylor’s sold the house in 1830 to the first Marquess Conyngham.

Taylor’s life growing up in the rarified environment of Bifrons House must have been like scenes out of the popular BBC TV drama Downton Abbey.  The house had a large staff of servants and large grounds at the edge of a large park near the town of Patrixbourne. Life as the heir to the estate would have been filled with social events and fine arts that included music and painting. Taylor developed a life-long love of music during his childhood, later collaborating with Isaac Newton on a scientific investigation of music (it was never published). He was also an amateur artist, and one of the first books he published after being elected to the Royal Society was on the mathematics of linear perspective, which contained some of the early results of projective geometry.

There is a beautiful family portrait in the National Portrait Gallery in London painted by John Closterman around 1696. The portrait is of the children of John Taylor about a year after he purchased Bifrons House. The painting is notable because Brook, the heir to the family fortunes, is being crowned with a wreath by his two older sisters (who would not inherit). Brook was only about 11 years old at the time and was already famous within his family for his ability with music and numbers.

Portrait of the children of John Taylor around 1696. Brook Taylor is the boy being crowned by his sisters on the left. (National Portrait Gallery)

Taylor never had to go to school, being completely tutored at home until he entered St. John’s College, Cambridge, in 1701.  He took mathematics classes from Machin and Keill and graduated in 1709.  The allowance from his father was sufficient to allow him to lead the life of a gentleman scholar, and he was elected a member of the Royal Society in 1712 and elected secretary of the Society just two years later.  During the following years he was active as a rising mathematician until 1721 when he married a woman of a good family but of no wealth.  The support of a house like Bifrons always took money, and the new wife’s lack of it was enough for Taylor’s father to throw the new couple out.  Unfortunately, his wife died in childbirth along with the child, so Taylor returned home in 1723.  These family troubles ended his main years of productivity as a mathematician.

Portrait of Brook Taylor

Methodus incrementorum directa et inversa

Under the eye of the Newtonian mathematician Keill at Cambridge, Taylor became a staunch supporter and user of Newton’s fluxions. Just after he was elected as a member of the Royal Society in 1712, he participated in an investigation of the priority for the invention of the calculus that pitted the British Newtonians against the Continental Leibnizians. The Royal Society found in favor of Newton (obviously) and raised the possibility that Leibniz learned of Newton’s ideas during a visit to England just a few years before Leibniz developed his own version of the differential calculus.

A re-evaluation of the priority dispute from today’s perspective attributes the calculus to both men. Newton clearly developed it first, but did not publish until much later. Leibniz published first and generated the excitement for the new method that dispersed its use widely. He also took an alternative route to the differential calculus that is demonstrably different than Newton’s. Did Leibniz benefit from possibly knowing Newton’s results (but not his methods)? Probably. But that is how science is supposed to work … building on the results of others while bringing new perspectives. Leibniz’ methods and his notations were superior to Newton’s, and the calculus we use today is closer to Leibniz’ version than to Newton’s.

Once Taylor was introduced to Newton’s fluxions, he latched on and helped push its development. The same year (1715) that he published a book on linear perspective for art, he also published a ground-breaking book on the use of the calculus to solve practical problems. This book, Methodus incrementorum directa et inversa, introduced several new ideas, including finite difference methods (which are used routinely today in numerical simulations of differential equations). It also considered possible solutions to the equation for a vibrating string for the first time.

The vibrating string is one of the simplest problem in “continuum mechanics”, but it posed a severe challenge to Newtonian physics of point particles. It was only much later that D’Alembert used Newton’s first law of action-reaction to eliminate internal forces to derive D’Alembert’s principle on the net force on an extended body. Yet Taylor used finite differences to treat the line mass of the string in a way that yielded a possible solution of a sine function. Taylor was the first to propose that a sine function was the form of the string displacement during vibration. This idea would be taken up later by D’Alembert (who first derived the wave equation), and by Euler (who vehemently disagreed with D’Alembert’s solutions) and Daniel Bernoulli (who was the first to suggest that it is not just a single sine function, but a sum of sine functions, that described the string’s motion — the principle of superposition).

Of course, the most influential idea in Taylor’s 1715 book was his general use of an infinite series to describe a curve.

Taylor’s Series

Infinite series became a major new tool in the toolbox of analysis with the publication of John WallisArithmetica Infinitorum published in 1656. Shortly afterwards many series were published such as Nikolaus Mercator‘s series (1668)

and James Gregory‘s series (1668)

And of course Isaac Newton’s generalized binomial theorem that he worked out famously during the plague years of 1665-1666

But these consisted mainly of special cases that had been worked out one by one. What was missing was a general method that could yield a series expression for any curve.

Taylor used concepts of finite differences as well as infinitesimals to derive his formula for expanding a function as a power series around any point. His derivation in Methodus incrementorum directa et inversa is not easily recognized today. Using difference tables, and ideas from Newton’s fluxions that viewed functions as curves traced out as a function of time, he arrived at the somewhat cryptic expression

where the “dots” are time derivatives, x stands for the ordinate (the function), v is a finite difference, and z is the abcissa moving with constant speed. If the abcissa moves with unit speed, then this becomes Taylor’s Series (in modern notation)

The term “Taylor’s series” was probably first used by L’Huillier in 1786, although Condorcet attributed the equation to both Taylor and d’Alembert in 1784. It was Lagrange in 1797 who immortalized Taylor by claiming that Taylor’s theorem was the foundation of analysis.

Example: sin(x)

Expand sin(x) around x = π

This is related to the expansion around x = 0 (also known as a Maclaurin series)

Example: arctan(x)

To get an feel for how to apply Taylor’s theorem to a function like arctan, begin with

and take the derivative of both sides

Rewrite this as

and substitute the expression for y

and integrate term by term to arrive at

This is James Gregory’s famous series. Although the math here is modern and only takes a few lines, it parallel’s Gregory’s approach. But Gregory had to invent aspects of calculus as he went along — his derivation covering many dense pages. In the priority dispute between Leibniz and Newton, Gregory is usually overlooked as an independent inventor of many aspects of the calculus. This is partly because Gregory acknowledged that Newton had invented it first, and he delayed publishing to give Newton priority.

Two-Dimensional Taylor’s Series

The ideas behind the Taylor’s series generalizes to any number of dimensions. For a scalar function of two variables it takes the form (out to second order)

where J is the Jacobian matrix (vector) and H is the Hessian matrix defined for the scalar function as


As a concrete example, consider the two-dimensional Gaussian function

The Jacobean and Hessian matrices are

which are the first- and second-order coefficients of the Taylor series.


[1] “A History of Bifrons House”, B. M. Thomas, Kent Archeological Society (2017)

[2] L. Feigenbaum, “TAYLOR,BROOK AND THE METHOD OF INCREMENTS,” Archive for History of Exact Sciences, vol. 34, no. 1-2, pp. 1-140, (1985)

[3] A. Malet, “GREGORIE, JAMES ON TANGENTS AND THE TAYLOR RULE FOR SERIES EXPANSIONS,” Archive for History of Exact Sciences, vol. 46, no. 2, pp. 97-137, (1993)

[4] E. Harier and G. Wanner, Analysis by its History (Springer, 1996)

Painting of Bifrons Park by Jan Wyck c. 1700

Johann Bernoulli’s Brachistochrone

Johann Bernoulli was an acknowledged genius–and he acknowledged it of himself.  Some flavor of his character can be seen in his opening lines of one of the most famous challenges in the history of mathematics—the statement of the Brachistrochrone Challenge.

“I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument.”

Of course, he meant his own fame, because he thought he already had a solution to the problem he posed to the mathematical community of the day.

The Problem of Fastest Descent

The problem posed by Johann Bernoulli was the brachistochrone (Gk: brachis + chronos) or the path of fastest descent. 

Galileo had attempted to tackle this problem in his Two New Sciences and had concluded, based on geometric arguments, that the solution was a circular path.  Yet he hedged—he confessed that he had reservations about this conclusion and suggested that a “higher mathematics” would possibly find a better solution. In fact he was right.

Fig. 1  Galileo considered a mass falling along different chords of a circle starting at A.  He proved that the path along ABG was quicker than along AG, and ABCG was quicker than ABG, and ABCDG was quicker than ABCG, etc.  In this way he showed that the path along the circular arc was quicker than any set of chords.  From this he inferred that the circle was the path of quickest descent—but he held out reservations, and rightly so.

In 1659, when Christiaan Huygens was immersed in the physics of pendula and time keeping, he was possibly the first mathematician to recognize that a perfect harmonic oscillator, one whose restoring force was linear in the displacement of the oscillator, would produce the perfect time piece.  Unfortunately, the pendulum, proposed by Galileo, was the simplest oscillator to construct, but Huygens already knew that it was not a perfect harmonic oscillator.  The period of oscillation became smaller when the amplitude of the oscillation became larger.  In order to “fix” the pendulum, he searched for a curve of equal time, called the tautochrone, that would allow all amplitudes of the pendulum to have the same period.  He found the solution and recognized it to be a cycloid arc. 

On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other…

His derivation filled 16 pages with geometric arguments, which was not a very efficient way to derive the thing.

Almost thirty years later, during the infancy of the infinitesimal calculus, the tautochrone was held up as a master example of an “optimal” solution whose derivation should yield to the much more powerful and elegant methods of the calculus.  Jakob Bernoulli, Johann’s brother, succeeded in deriving the tautochrone in 1690 using the calculus, using the term “integral” for the first time in print, but it was not at first clear what other problems could yield in a similar way.

Then, in 1696, Johann Bernoulli posed the brachistrochrone problem in the pages of Acta Eruditorum.

Fig. 2 The shortest-time route from A to B, relying only on gravity, is the cycloid, compared to the parabola, circle and linear paths. Johann and Jakob Bernoulli, brothers, competed to find the best solution.

Acta Eruditorum

The Acta Eruditorum was the German answer to the Proceedings of the Royal Society of London.  It began publishing in Leipzig in 1682 under the editor Otto Mencke.  Although Mencke was the originator, launching and supporting the journal became the obsession of Gottfried Lebiniz, who felt he was a hostage in the backwaters of Hanover Germany but who yearned for a place on the world stage (i.e. Paris or London).  By launching the continental publication, the Continental scientists had a freer voice without needing to please the gate keepers at the Royal Society.  And by launching a German journal, it gave German scientists like Leibniz (and the Bernoullis and Euler, and von Tschirnhaus among others) a freer voice without censor by the Journal des Savants of Paris.

Fig. 3 Acta Eruditorum of 1684 containing one of Leibniz’ early papers on the calculus.

The Acta Eruditorum was almost a vanity press for Leibniz.  He published 13 papers in the journal in its first 4 years of activity starting in 1682.  In return, when Leibniz became embroiled in the priority dispute with Newton over the invention of the calculus, the Acta provided loyal support for Leibniz’ side just as the Proceedings of the Royal Society gave loyal support to Newton.  In fact, the trigger that launched the nasty battle with Newton was a review that Leibniz wrote for the Acta in 17?? [Ref] in which he presented himself as the primary inventor of the calculus.  When he failed to give due credit, not only to Newton, but also to lesser contributors, they fought back by claiming that Leibniz had stolen the idea from Newton.  Although a kangaroo court by the Royal Society found in favor of Newton, posterity gives most of the credit for the development and dissemination of the calculus to Leibniz.  Where Newton guarded his advances jealously and would not explain his approach, Leibniz freely published his methods for all to see and to learn and to try out for themselves.  In this open process, the Acta was the primary medium of communication and gets the credit for being the conduit by which the calculus was presented to the world.

Although the Acta Eruditorum only operated for 100 years, it stands out as the most important publication for the development of the calculus.  Leibnitz published in the Acta a progressive set of papers that outlined his method for the calculus.  More importantly, his papers elicited responses from other mathematicians, most notably Johann Bernoulli and von Tschirnhaus and L’Hopital, who helped to refine the methods and advance the art.  The Acta became a collaborative space for this team of mathematicians as they fine-tuned the methods as well as the notations for the calculus, most of which stand to this day.  In contrast, Newton’s notations have all but faded, save the simple “dot” notation over variables to denote them as time derivatives (his fluxions).  Therefore, for most of continental Europe, the Acta Eruditorum was the place to publish, and it was here that Johann Bernoulli published his famous challenge of the brachistochrone.

The Competition

Johann suggested the problem in the June 1696 Acta Eruditorum

Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time

The competition was originally proposed for 6 months, but it was then extended to a year and a half.  Johann published his results about a year later, but not without controversy.  Johann had known that his brother Jakob was also working on the problem, but he incorrectly thought that Jakob was convinced that Galileo had been right, so Johann described his approach to Jakob thinking he had little to fear in the competition.  Johann didn’t know that Jakob had already taken an approach similar to Johann’s, and even more importantly, Jakob had done the math correctly.  When Jakob showed Johann his mistake, he also ill-advisedly showed him the correct derivation.  Johann sent off a manuscript to Acta with the correct derivation that he had learned from Jakob.

Within the year and a half there were 4 additional solutions—all correct—using different approaches.  One of the most famous responses was by Newton (who as usual did not give up his method) but who is reported to have solved the problem in a day.  Others who contributed solutions were Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l’Hôpital’s.  Of course, Jakob sent in his own solution, although it overlapped with the one Johann had already published.

The Solution of Jakob and Johann Bernoulli

The stroke of genius of Jakob and Johann Bernoulli, accomplished in 1697 only about 20 years after the invention of the calculus, was to recognize an amazing analogy between mechanics and light.  Their insight foreshadowed Lagrange by a hundred years and William Rowan Hamilton by a hundred and fifty.  They did this by recognizing that the path of a light beam, just like the trajectory of a particle, conserves certain properties.  In the case of Fermat’s principle, a light ray refracts to take the path of least time between two points.  The insight of the Bernoulli’s is that a mechanical particle would behave in exactly the same way.  Therefore, the brachistrochrone can be obtained by considering the path that a light beam would take if the light ray were propagating through a medium with non-uniform refractive index to that the speed of light varies with height y as

Fermat’s principle of least time, which is consistent with Snell’s Law at interfaces, imposes the constraint on the path

This equation for a light ray propagating through a non-uniform medium would later become known as the Eikonal Equation.  The conserved quantity along this path is the value 1/vm.  Rewriting the Eikonal equation as

it can be solved for the differential equation

which those in the know (as certainly the Bernoullis were) would know is the equation of a cycloid.  If the sliding bead is on a wire shaped like a cycloid, there must be a lowest point for which the speed is a maximum.  For the cycloid curve of diameter D, this is

Therefore, the equation for the brachistochrone is

which is the differential equation for an inverted cycloid of diameter D.

Fig. 4 A light ray enters vertically on a medium whose refractive index varies as the square-root of depth.  The path of least time for the light ray to travel through the material is a cycloid—the same as for a massive particle traveling from point A to point B.

Calculus of Variations

Variational calculus had not quite been invented in time to solve the Brachistochrone, although the brachistochrone challenge helped motivate its eventual development by Euler and Lagrange later in the eighteenth century. Nonetheless, it is helpful to see the variational solution, which is the way we would solve this problem if it were a Lagrangian problem in advanced classical mechanics.

First, the total time taken by the sliding bead is defined as

Then we take energy conservation to solve for v(y)

The path element is

which leads to the expression for total time

It is the argument of the integral which is the quantity to be varied (the Lagrangian)

which can be inserted into the Lagrange equation

This has a simple first integral

This is explicitly solved

Once again, it helps to recognize the equation of a cycloid, because the last line can be solved as the parametric curves

which is the cycloid curve.


C. B. Boyer, The History of the Calculus and its Conceptual Development. New York: Dover, 1959.

J. Coopersmith, The lazy universe : an introduction to the principle of least action. Oxford University Press, 2017.

D. S. Lemons, Perfect Form: Variational Principles, Methods, and Applications in Elementary Physics. Princeton University Press, 1997.

Wikipedia: The Brachistrochrone Curve

W. Yourgrau, Variational principles in dynamics and quantum theory, 2d ed.. ed. New York: New York, Pitman Pub. Corp., 1960.

The Solvay Debates: Einstein versus Bohr

Einstein is the alpha of the quantum. Einstein is also the omega. Although he was the one who established the quantum of energy and matter (see my Blog Einstein vs Planck), Einstein pitted himself in a running debate against Niels Bohr’s emerging interpretation of quantum physics that had, in Einstein’s opinion, severe deficiencies. Between sessions during a series of conferences known as the Solvay Congresses over a period of eight years from 1927 to 1935, Einstein constructed a challenges of increasing sophistication to confront Bohr and his quasi-voodoo attitudes about wave-function collapse. To meet the challenge, Bohr sharpened his arguments and bested Einstein, who ultimately withdrew from the field of battle. Einstein, as quantum physics’ harshest critic, played a pivotal role, almost against his will, establishing the Copenhagen interpretation of quantum physics that rules to this day, and also inventing the principle of entanglement which lies at the core of almost all quantum information technology today.

Debate Timeline

  • Fifth Solvay Congress: 1927 October Brussels: Debate Round 1
    • Einstein and ensembles
  • Sixth Solvay Congress: 1930 Debate Round 2
    • Photon in a box
  • Seventh Solvay Congress: 1933
    • Einstein absent (visiting the US when Hitler takes power…decides not to return to Germany.)
  • Physical Review 1935: Debate Round 3
    • EPR paper and Bohr’s response
    • Schrödinger’s Cat
  • Notable Nobel Prizes
    • 1918 Planck
    • 1921 Einstein
    • 1922 Bohr
    • 1932 Heisenberg
    • 1933 Dirac and Schrödinger

The Solvay Conferences

The Solvay congresses were unparalleled scientific meetings of their day.  They were attended by invitation only, and invitations were offered only to the top physicists concerned with the selected topic of each meeting.  The Solvay congresses were held about every three years always in Belgium, supported by the Belgian chemical industrialist Ernest Solvay.  The first meeting, held in 1911, was on the topic of radiation and quanta. 

Fig. 1 First Solvay Congress (1911). Einstein (standing second from right) was one of the youngest attendees.

The fifth meeting, held in 1927, was on electrons and photons and focused on the recent rapid advances in quantum theory.  The old quantum guard was invited—Planck, Bohr and Einstein.  The new quantum guard was invited as well—Heisenberg, de Broglie, Schrödinger, Born, Pauli, and Dirac.  Heisenberg and Bohr joined forces to present a united front meant to solidify what later became known as the Copenhagen interpretation of quantum physics.  The basic principles of the interpretation include the wavefunction of Schrödinger, the probabilistic interpretation of Born, the uncertainty principle of Heisenberg, the complementarity principle of Bohr and the collapse of the wavefunction during measurement.  The chief conclusion that Heisenberg and Bohr sought to impress on the assembled attendees was that the theory of quantum processes was complete, meaning that unknown or uncertain  characteristics of measurements could not be attributed to lack of knowledge or understanding, but were fundamental and permanently inaccessible.

Fig. 2 Fifth Solvay Congress (1927). Einstein front and center. Bohr on the far right middle row.

Einstein was not convinced with that argument, and he rose to his feet to object after Bohr’s informal presentation of his complementarity principle.  Einstein insisted that uncertainties in measurement were not fundamental, but were caused by incomplete information, that , if known, would accurately account for the measurement results.  Bohr was not prepared for Einstein’s critique and brushed it off, but what ensued in the dining hall and the hallways of the Hotel Metropole in Brussels over the next several days has become one of the most famous scientific debates of the modern era, known as the Bohr-Einstein debate on the meaning of quantum theory.  The debate gently raged night and day through the fifth congress, and was renewed three years later at the 1930 congress.  It finished, in a final flurry of published papers in 1935 that launched some of the central concepts of quantum theory, including the idea of quantum entanglement and, of course, Schrödinger’s cat.

Einstein’s strategy, to refute Bohr, was to construct careful thought experiments that envisioned perfect experiments, without errors, that measured properties of ideal quantum systems.  His aim was to paint Bohr into a corner from which he could not escape, caught by what Einstein assumed was the inconsistency of complementarity.  Einstein’s “thought experiments” used electrons passing through slits, diffracting as required by Schrödinger’s theory, but being detected by classical measurements.  Einstein would present a thought experiment to Bohr, who would then retreat to consider the way around Einstein’s arguments, returning the next hour or the next day with his answer, only to be confronted by yet another clever device of Einstein’s clever imagination that would force Bohr to retreat again.  The spirit of this back and forth encounter between Bohr and Einstein is caught dramatically in the words of Paul Ehrenfest who witnessed the debate first hand, partially mediating between Bohr and Einstein, both of whom he respected deeply.

“Brussels-Solvay was fine!… BOHR towering over everybody.  At first not understood at all … , then  step by step defeating everybody.  Naturally, once again the awful Bohr incantation terminology.  Impossible for anyone else to summarise … (Every night at 1 a.m., Bohr came into my room just to say ONE SINGLE WORD to me, until three a.m.)  It was delightful for me to be present during the conversation between Bohr and Einstein.  Like a game of chess, Einstein all the time with new examples.  In a certain sense a sort of Perpetuum Mobile of the second kind to break the UNCERTAINTY RELATION.  Bohr from out of philosophical smoke clouds constantly searching for the tools to crush one example after the other.  Einstein like a jack-in-the-box; jumping out fresh every morning.  Oh, that was priceless.  But I am almost without reservation pro Bohr and contra Einstein.  His attitude to Bohr is now exacly like the attitude of the defenders of absolute simultaneity towards him …” [1]

The most difficult example that Einstein constructed during the fifth Solvary Congress involved an electron double-slit apparatus that could measure, in principle, the momentum imparted to the slit by the passing electron, as shown in Fig.3.  The electron gun is a point source that emits the electrons in a range of angles that illuminates the two slits.  The slits are small relative to a de Broglie wavelength, so the electron wavefunctions diffract according to Schrödinger’s wave mechanics to illuminate the detection plate.  Because of the interference of the electron waves from the two slits, electrons are detected clustered in intense fringes separated by dark fringes. 

So far, everyone was in agreement with these suggested results.  The key next step is the assumption that the electron gun emits only a single electron at a time, so that only one electron is present in the system at any given time.  Furthermore, the screen with the double slit is suspended on a spring, and the position of the screen is measured with complete accuracy by a displacement meter.  When the single electron passes through the entire system, it imparts a momentum kick to the screen, which is measured by the meter.  It is also detected at a specific location on the detection plate.  Knowing the position of the electron detection, and the momentum kick to the screen, provides information about which slit the electron passed through, and gives simultaneous position and momentum values to the electron that have no uncertainty, apparently rebutting the uncertainty principle.             

Fig. 3 Einstein’s single-electron thought experiment in which the recoil of the screen holding the slits can be measured to tell which way the electron went. Bohr showed that the more “which way” information is obtained, the more washed-out the interference pattern becomes.

This challenge by Einstein was the culmination of successively more sophisticated examples that he had to pose to combat Bohr, and Bohr was not going to let it pass unanswered.  With ingenious insight, Bohr recognized that the key element in the apparatus was the fact that the screen with the slits must have finite mass if the momentum kick by the electron were to produce a measurable displacement.  But if the screen has finite mass, and hence a finite momentum kick from the electron, then there must be an uncertainty in the position of the slits.  This uncertainty immediately translates into a washout of the interference fringes.  In fact the more information that is obtained about which slit the electron passed through, the more the interference is washed out.  It was a perfect example of Bohr’s own complementarity principle.  The more the apparatus measures particle properties, the less it measures wave properties, and vice versa, in a perfect balance between waves and particles. 

Einstein grudgingly admitted defeat at the end of the first round, but he was not defeated.  Three years later he came back armed with more clever thought experiments, ready for the second round in the debate.

The Sixth Solvay Conference: 1930

At the Solvay Congress of 1930, Einstein was ready with even more difficult challenges.  His ultimate idea was to construct a box containing photons, just like the original black bodies that launched Planck’s quantum hypothesis thirty years before.  The box is attached to a weighing scale so that the weight of the box plus the photons inside can be measured with arbitrarily accuracy. A shutter over a hole in the box is opened for a time T, and a photon is emitted.  Because the photon has energy, it has an equivalent weight (Einstein’s own famous E = mc2), and the mass of the box changes by an amount equal to the photon energy divided by the speed of light squared: m = E/c2.  If the scale has arbitrary accuracy, then the energy of the photon has no uncertainty.  In addition, because the shutter was open for only a time T, the time of emission similarly has no uncertainty.  Therefore, the product of the energy uncertainty and the time uncertainty is much smaller than Planck’s constant, apparently violating Heisenberg’s precious uncertainty principle.

Bohr was stopped in his tracks with this challenge.  Although he sensed immediately that Einstein had missed something (because Bohr had complete confidence in the uncertainty principle), he could not put his finger immediately on what it was.  That evening he wandered from one attendee to another, very unhappy, trying to persuade them and saying that Einstein could not be right because it would be the end of physics.  At the end of the evening, Bohr was no closer to a solution, and Einstein was looking smug.  However, by the next morning Bohr reappeared tired but in high spirits, and he delivered a master stroke.  Where Einstein had used special relaitivity against Bohr, Bohr now used Einstein’s own general relativity against him. 

The key insight was that the weight of the box must be measured, and the process of measurement was just as important as the quantum process being measured—this was one of the cornerstones of the Copenhagen interpretation.  So Bohr envisioned a measuring apparatus composed of a spring and a scale with the box suspended in gravity from the spring.  As the photon leaves the box, the weight of the box changes, and so does the deflection of the spring, changing the height of the box.  This change in height, in a gravitational potential, causes the timing of the shutter to change according to the law of gravitational time dilation in general relativity.  By calculating the the general relativistic uncertainty in the time, coupled with the special relativistic uncertainty in the weight of the box, produced a product that was at least as big as Planck’s constant—Heisenberg’s uncertainty principle was saved!

Fig. 4 Einstein’s thought experiment that uses special relativity to refute quantum mechanics. Bohr then invoked Einstein’s own general relativity to refute him.

Entanglement and Schrödinger’s Cat

Einstein ceded the point to Bohr but was not convinced. He still believed that quantum mechanics was not a “complete” theory of quantum physics and he continued to search for the perfect thought experiment that Bohr could not escape. Even today when we have become so familiar with quantum phenomena, the Copenhagen interpretation of quantum mechanics has weird consequences that seem to defy common sense, so it is understandable that Einstein had his reservations.

After the sixth Solvay congress Einstein and Schrödinger exchanged many letters complaining to each other about Bohr’s increasing strangle-hold on the interpretation of quantum mechanics. Egging each other on, they both constructed their own final assault on Bohr. The irony is that the concepts they devised to throw down quantum mechanics have today become cornerstones of the theory. For Einstein, his final salvo was “Entanglement”. For Schrödinger, his final salvo was his “cat”. Today, Entanglement and Schrödinger’s Cat have become enshrined on the alter of quantum interpretation even though their original function was to thwart that interpretation.

The final round of the debate was carried out, not at a Solvay congress, but in the Physical review journal by Einstein [2] and Bohr [3], and in the Naturwissenshaften by Schrödinger [4].

In 1969, Heisenberg looked back on these years and said,

To those of us who participated in the development of atomic theory, the five years following the Solvay Conference in Brussels in 1927 looked so wonderful that we often spoke of them as the golden age of atomic physics. The great obstacles that had occupied all our efforts in the preceding years had been cleared out of the way, the gate to an entirely new field, the quantum mechanics of the atomic shells stood wide open, and fresh fruits seemed ready for the picking. [5]


[1] A. Whitaker, Einstein, Bohr, and the quantum dilemma : from quantum theory to quantum information, 2nd ed. Cambridge University Press, 2006. (pg. 210)

[2] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 47, no. 10, pp. 0777-0780, May (1935)

[3] N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 48, no. 8, pp. 696-702, Oct (1935)

[4] E. Schrodinger, “The current situation in quantum mechanics,” Naturwissenschaften, vol. 23, pp. 807-812, (1935)

[5] W Heisenberg, Physics and beyond : Encounters and conversations (Harper, New York, 1971)

Timelines in the History and Physics of Dynamics (with links to primary texts)

These timelines in the History of Dynamics are organized along the Chapters in Galileo Unbound (Oxford, 2018). The book is about the physics and history of dynamics including classical and quantum mechanics as well as general relativity and nonlinear dynamics (with a detour down evolutionary dynamics and game theory along the way). The first few chapters focus on Galileo, while the following chapters follow his legacy, as theories of motion became more abstract, eventually to encompass the evolution of species within the same theoretical framework as the orbit of photons around black holes.

Galileo: A New Scientist

Galileo Galilei was the first modern scientist, launching a new scientific method that superseded, after one and a half millennia, Aristotle’s physics.  Galileo’s career began with his studies of motion at the University of Pisa that were interrupted by his move to the University of Padua and his telescopic discoveries of mountains on the moon and the moons of Jupiter.  Galileo became the first rock star of science, and he used his fame to promote the ideas of Copernicus and the Sun-centered model of the solar system.  But he pushed too far when he lampooned the Pope.  Ironically, Galileo’s conviction for heresy and his sentence to house arrest for the remainder of his life gave him the free time to finally finish his work on the physics of motion, which he published in Two New Sciences in 1638.

1543 Copernicus dies, publishes posthumously De Revolutionibus

1564    Galileo born

1581    Enters University of Pisa

1585    Leaves Pisa without a degree

1586    Invents hydrostatic balance

1588    Receives lecturship in mathematics at Pisa

1592    Chair of mathematics at Univeristy of Padua

1595    Theory of the tides

1595    Invents military and geometric compass

1596    Le Meccaniche and the principle of horizontal inertia

1600    Bruno Giordano burned at the stake

1601    Death of Tycho Brahe

1609    Galileo constructs his first telescope, makes observations of the moon

1610    Galileo discovers 4 moons of Jupiter, Starry Messenger (Sidereus Nuncius), appointed chief philosopher and mathematician of the Duke of Tuscany, moves to Florence, observes Saturn, Venus goes through phases like the moon

1611    Galileo travels to Rome, inducted into the Lyncean Academy, name “telescope” is first used

1611    Scheiner discovers sunspots

1611    Galileo meets Barberini, a cardinal

1611 Johannes Kepler, Dioptrice

1613    Letters on sunspots published by Lincean Academy in Rome

1614    Galileo denounced from the pulpit

1615    (April) Bellarmine writes an essay against Coperinicus

1615    Galileo investigated by the Inquisition

1615    Writes Letter to Christina, but does not publish it

1615    (December) travels to Rome and stays at Tuscan embassy

1616    (January) Francesco Ingoli publishes essay against Copernicus

1616    (March) Decree against copernicanism

1616    Galileo publishes theory of tides, Galileo meets with Pope Paul V, Copernicus’ book is banned, Galileo warned not to support the Coperinican system, Galileo decides not to reply to Ingoli, Galileo proposes eclipses of Jupter’s moons to determine longitude at sea

1618    Three comets appear, Grassi gives a lecture not hostile to Galileo

1618    Galileo, through Mario Guiducci, publishes scathing attack on Grassi

1619    Jesuit Grassi (Sarsi) publishes attack on Galileo concerning 3 comets

1619    Marina Gamba dies, Galileo legitimizes his son Vinczenzio

1619 Kepler’s Laws, Epitome astronomiae Copernicanae.

1623    Barberini becomes Urban VIII, The Assayer published (response to Grassi)

1624    Galileo visits Rome and Urban VIII

1629    Birth of his grandson Galileo

1630    Death of Johanes Kepler

1632    Publication of the Dialogue Concerning the Two Chief World Systems, Galileo is indicted by the Inquisition (68 years old)

1633    (February) Travels to Rome

1633    Convicted, abjurs, house arrest in Rome, then Siena, then home to Arcetri

1638    Blind, publication of Two New Sciences

1642    Galileo dies (77 years old)

Galileo’s Trajectory

Galileo’s discovery of the law of fall and the parabolic trajectory began with early work on the physics of motion by predecessors like the Oxford Scholars, Tartaglia and the polymath Simon Stevin who dropped lead weights from the leaning tower of Delft three years before Galileo (may have) dropped lead weights from the leaning tower of Pisa.  The story of how Galileo developed his ideas of motion is described in the context of his studies of balls rolling on inclined plane and the surprising accuracy he achieved without access to modern timekeeping.

1583    Galileo Notices isochronism of the pendulum

1588    Receives lecturship in mathematics at Pisa

1589 – 1592  Work on projectile motion in Pisa

1592    Chair of mathematics at Univeristy of Padua

1596    Le Meccaniche and the principle of horizontal inertia

1600    Guidobaldo shares technique of colored ball

1602    Proves isochronism of the pendulum (experimentally)

1604    First experiments on uniformly accelerated motion

1604    Wrote to Scarpi about the law of fall (s ≈ t2)

1607-1608  Identified trajectory as parabolic

1609    Velocity proportional to time

1632    Publication of the Dialogue Concerning the Two Chief World Systems, Galileo is indicted by the Inquisition (68 years old)

1636    Letter to Christina published in Augsburg in Latin and Italian

1638    Blind, publication of Two New Sciences

1641    Invented pendulum clock (in theory)

1642    Dies (77 years old)

On the Shoulders of Giants

Galileo’s parabolic trajectory launched a new approach to physics that was taken up by a new generation of scientists like Isaac Newton, Robert Hooke and Edmund Halley.  The English Newtonian tradition was adopted by ambitious French iconoclasts who championed Newton over their own Descartes.  Chief among these was Pierre Maupertuis, whose principle of least action was developed by Leonhard Euler and Joseph Lagrange into a rigorous new science of dynamics.  Along the way, Maupertuis became embroiled in a famous dispute that entangled the King of Prussia as well as the volatile Voltaire who was mourning the death of his mistress Emilie du Chatelet, the lone female French physicist of the eighteenth century.

1644    Descartes’ vortex theory of gravitation

1662    Fermat’s principle

1669 – 1690    Huygens expands on Descartes’ vortex theory

1687 Newton’s Principia

1698    Maupertuis born

1729    Maupertuis entered University in Basel.  Studied under Johann Bernoulli

1736    Euler publishes Mechanica sive motus scientia analytice exposita

1737   Maupertuis report on expedition to Lapland.  Earth is oblate.  Attacks Cassini.

1744    Maupertuis Principle of Least Action.  Euler Principle of Least Action.

1745    Maupertuis becomes president of Berlin Academy.  Paris Academy cancels his membership after a campaign against him by Cassini.

1746    Maupertuis principle of Least Action for mass

1751    Samuel König disputes Maupertuis’ priority

1756    Cassini dies.  Maupertuis reinstated in the French Academy

1759    Maupertuis dies

1759    du Chatelet’s French translation of Newton’s Principia published posthumously

1760    Euler 3-body problem (two fixed centers and coplanar third body)

1760-1761 Lagrange, Variational calculus (J. L. Lagrange, “Essai d’une nouvelle méthod pour dEeterminer les maxima et lest minima des formules intégrales indéfinies,” Miscellanea Teurinensia, (1760-1761))

1762    Beginning of the reign of Catherine the Great of Russia

1763    Euler colinear 3-body problem

1765    Euler publishes Theoria motus corporum solidorum on rotational mechanics

1766    Euler returns to St. Petersburg

1766    Lagrange arrives in Berlin

1772    Lagrange equilateral 3-body problem, Essai sur le problème des trois corps, 1772, Oeuvres tome 6

1775    Beginning of the American War of Independence

1776    Adam Smith Wealth of Nations

1781    William Herschel discovers Uranus

1783    Euler dies in St. Petersburg

1787    United States Constitution written

1787    Lagrange moves from Berlin to Paris

1788    Lagrange, Méchanique analytique

1789    Beginning of the French Revolution

1799    Pierre-Simon Laplace Mécanique Céleste (1799-1825)

Geometry on My Mind

This history of modern geometry focuses on the topics that provided the foundation for the new visualization of physics.  It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics.  Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics.  Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory.

1629    Fermat described higher-dim loci

1637    Descarte’s Geometry

1649    van Schooten’s commentary on Descartes Geometry

1694    Leibniz uses word “coordinate” in its modern usage

1697    Johann Bernoulli shortest distance between two points on convex surface

1732    Euler geodesic equations for implicit surfaces

1748    Euler defines modern usage of function

1801    Gauss calculates orbit of Ceres

1807    Fourier analysis (published in 1822(

1807    Gauss arrives in Göttingen

1827    Karl Gauss establishes differential geometry of curved surfaces, Disquisitiones generales circa superficies curvas

1830    Bolyai and Lobachevsky publish on hyperbolic geometry

1834    Jacobi n-fold integrals and volumes of n-dim spheres

1836    Liouville-Sturm theorem

1838    Liouville’s theorem

1841    Jacobi determinants

1843    Arthur Cayley systems of n-variables

1843    Hamilton discovers quaternions

1844    Hermann Grassman n-dim vector spaces, Die Lineale Ausdehnungslehr

1846    Julius Plücker System der Geometrie des Raumes in neuer analytischer Behandlungsweise

1848 Jacobi Vorlesungen über Dynamik

1848    “Vector” coined by Hamilton

1854    Riemann’s habilitation lecture

1861    Riemann n-dim solution of heat conduction

1868    Publication of Riemann’s Habilitation

1869    Christoffel and Lipschitz work on multiple dimensional analysis

1871    Betti refers to the n-ply of numbers as a “space”.

1871    Klein publishes on non-euclidean geometry

1872 Boltzmann distribution

1872    Jordan Essay on the geometry of n-dimensions

1872    Felix Klein’s “Erlangen Programme”

1872    Weierstrass’ Monster

1872    Dedekind cut

1872    Cantor paper on irrational numbers

1872    Cantor meets Dedekind

1872 Lipschitz derives mechanical motion as a geodesic on a manifold

1874    Cantor beginning of set theory

1877    Cantor one-to-one correspondence between the line and n-dimensional space

1881    Gibbs codifies vector analysis

1883    Cantor set and staircase Grundlagen einer allgemeinen Mannigfaltigkeitslehre

1884    Abbott publishes Flatland

1887    Peano vector methods in differential geometry

1890    Peano space filling curve

1891    Hilbert space filling curve

1887    Darboux vol. 2 treats dynamics as a point in d-dimensional space.  Applies concepts of geodesics for trajectories.

1898    Ricci-Curbastro Lesons on the Theory of Surfaces

1902    Lebesgue integral

1904    Hilbert studies integral equations

1904    von Koch snowflake

1906    Frechet thesis on square summable sequences as infinite dimensional space

1908    Schmidt Geometry in a Function Space

1910    Brouwer proof of dimensional invariance

1913    Hilbert space named by Riesz

1914    Hilbert space used by Hausdorff

1915    Sierpinski fractal triangle

1918    Hausdorff non-integer dimensions

1918    Weyl’s book Space, Time, Matter

1918    Fatou and Julia fractals

1920    Banach space

1927    von Neumann axiomatic form of Hilbert Space

1935    Frechet full form of Hilbert Space

1967    Mandelbrot coast of Britain

1982    Mandelbrot’s book The Fractal Geometry of Nature

The Tangled Tale of Phase Space

Phase space is the central visualization tool used today to study complex systems.  The chapter describes the origins of phase space with the work of Joseph Liouville and Carl Jacobi that was later refined by Ludwig Boltzmann and Rudolf Clausius in their attempts to define and explain the subtle concept of entropy.  The turning point in the history of phase space was when Henri Poincaré used phase space to solve the three-body problem, uncovering chaotic behavior in his quest to answer questions on the stability of the solar system.  Phase space was established as the central paradigm of statistical mechanics by JW Gibbs and Paul Ehrenfest.

1804    Jacobi born (1904 – 1851) in Potsdam

1804    Napoleon I Emperor of France

1806    William Rowan Hamilton born (1805 – 1865)

1807    Thomas Young describes “Energy” in his Course on Natural Philosophy (Vol. 1 and Vol. 2)

1808    Bethoven performs his Fifth Symphony

1809    Joseph Liouville born (1809 – 1882)

1821    Hermann Ludwig Ferdinand von Helmholtz born (1821 – 1894)

1824    Carnot published Reflections on the Motive Power of Fire

1834    Jacobi n-fold integrals and volumes of n-dim spheres

1834-1835       Hamilton publishes his principle (1834, 1835).

1836    Liouville-Sturm theorem

1837    Queen Victoria begins her reign as Queen of England

1838    Liouville develops his theorem on products of n differentials satisfying certain first-order differential equations.  This becomes the classic reference to Liouville’s Theorem.

1847    Helmholtz  Conservation of Energy (force)

1849    Thomson makes first use of “Energy” (From reading Thomas Young’s lecture notes)

1850    Clausius establishes First law of Thermodynamics: Internal energy. Second law:  Heat cannot flow unaided from cold to hot.  Not explicitly stated as first and second laws

1851    Thomson names Clausius’ First and Second laws of Thermodynamics

1852    Thomson describes general dissipation of the universe (“energy” used in title)

1854    Thomson defined absolute temperature.  First mathematical statement of 2nd law.  Restricted to reversible processes

1854    Clausius stated Second Law of Thermodynamics as inequality

1857    Clausius constructs kinetic theory, Mean molecular speeds

1858    Clausius defines mean free path, Molecules have finite size. Clausius assumed that all molecules had the same speed

1860    Maxwell publishes first paper on kinetic theory. Distribution of speeds. Derivation of gas transport properties

1865    Loschmidt size of molecules

1865    Clausius names entropy

1868    Boltzmann adds (Boltzmann) factor to Maxwell distribution

1872    Boltzmann transport equation and H-theorem

1876    Loschmidt reversibility paradox

1877    Boltzmann  S = k logW

1890    Poincare: Recurrence Theorem. Recurrence paradox with Second Law (1893)

1896    Zermelo criticizes Boltzmann

1896    Boltzmann posits direction of time to save his H-theorem

1898    Boltzmann Vorlesungen über Gas Theorie

1905    Boltzmann kinetic theory of matter in Encyklopädie der mathematischen Wissenschaften

1906    Boltzmann dies

1910    Paul Hertz uses “Phase Space” (Phasenraum)

1911    Ehrenfest’s article in Encyklopädie der mathematischen Wissenschaften

1913    A. Rosenthal writes the first paper using the phrase “phasenraum”, combining the work of Boltzmann and Poincaré. “Beweis der Unmöglichkeit ergodischer Gassysteme” (Ann. D. Physik, 42, 796 (1913)

1913    Plancheral, “Beweis der Unmöglichkeit ergodischer mechanischer Systeme” (Ann. D. Physik, 42, 1061 (1913).  Also uses “Phasenraum”.

The Lens of Gravity

Gravity provided the backdrop for one of the most important paradigm shifts in the history of physics.  Prior to Albert Einstein’s general theory of relativity, trajectories were paths described by geometry.  After the theory of general relativity, trajectories are paths caused by geometry.  This chapter explains how Einstein arrived at his theory of gravity, relying on the space-time geometry of Hermann Minkowski, whose work he had originally harshly criticized.  The confirmation of Einstein’s theory was one of the dramatic high points in 20th century history of physics when Arthur Eddington journeyed to an island off the coast of Africa to observe stellar deflections during a solar eclipse.  If Galileo was the first rock star of physics, then Einstein was the first worldwide rock star of science.

1697    Johann Bernoulli was first to find solution to shortest path between two points on a curved surface (1697).

1728    Euler found the geodesic equation.

1783    The pair 40 Eridani B/C was discovered by William Herschel on 31 January

1783    John Michell explains infalling object would travel faster than speed of light

1796    Laplace describes “dark stars” in Exposition du system du Monde

1827    The first orbit of a binary star computed by Félix Savary for the orbit of Xi Ursae Majoris.

1827    Gauss curvature Theoriem Egregum

1844    Bessel notices periodic displacement of Sirius with period of half a century

1844    The name “geodesic line” is attributed to Liouville.

1845    Buys Ballot used musicians with absolute pitch for the first experimental verification of the Doppler effect

1854    Riemann’s habilitationsschrift

1862    Discovery of Sirius B (a white dwarf)

1868    Darboux suggested motions in n-dimensions

1872    Lipshitz first to apply Riemannian geometry to the principle of least action.

1895    Hilbert arrives in Göttingen

1902    Minkowski arrives in Göttingen

1905    Einstein’s miracle year

1906    Poincaré describes Lorentz transformations as rotations in 4D

1907    Einstein has “happiest thought” in November

1907    Einstein’s relativity review in Jahrbuch

1908    Minkowski’s Space and Time lecture

1908    Einstein appointed to unpaid position at University of Bern

1909    Minkowski dies

1909    Einstein appointed associate professor of theoretical physics at U of Zürich

1910    40 Eridani B was discobered to be of spectral type A (white dwarf)

1910    Size and mass of Sirius B determined (heavy and small)

1911    Laue publishes first textbook on relativity theory

1911    Einstein accepts position at Prague

1911    Einstein goes to the limits of special relativity applied to gravitational fields

1912    Einstein’s two papers establish a scalar field theory of gravitation

1912    Einstein moves from Prague to ETH in Zürich in fall.  Begins collaboration with Grossmann.

1913    Einstein EG paper

1914    Adams publishes spectrum of 40 Eridani B

1915    Sirius B determined to be also a low-luminosity type A white dwarf

1915    Einstein Completes paper

1916    Density of 40 Eridani B by Ernst Öpik

1916    Schwarzschild paper

1916 Einstein’s publishes theory of gravitational waves

1919    Eddington expedition to Principe

1920    Eddington paper on deflection of light by the sun

1922    Willem Luyten coins phrase “white dwarf”

1924    Eddington found a set of coordinates that eliminated the singularity at the Schwarzschild radius

1926    R. H. Fowler publishes paper on degenerate matter and composition of white dwarfs

1931    Chandrasekhar calculated the limit for collapse to white dwarf stars at 1.4MS

1933    Georges Lemaitre states the coordinate singularity was an artefact

1934    Walter Baade and Fritz Zwicky proposed the existence of the neutron star only a year after the discovery of the neutron by Sir James Chadwick.

1939    Oppenheimer and Snyder showed ultimate collapse of a 3MS  “frozen star”

1958    David Finkelstein paper

1965    Antony Hewish and Samuel Okoye discovered “an unusual source of high radio brightness temperature in the Crab Nebula”. This source turned out to be the Crab Nebula neutron star that resulted from the great supernova of 1054.

1967    Jocelyn Bell and Antony Hewish discovered regular radio pulses from CP 1919. This pulsar was later interpreted as an isolated, rotating neutron star.

1967    Wheeler’s “black hole” talk

1974    Joseph Taylor and Russell Hulse discovered the first binary pulsar, PSR B1913+16, which consists of two neutron stars (one seen as a pulsar) orbiting around their center of mass.

2015    LIGO detects gravitational waves on Sept. 14 from the merger of two black holes

2017    LIGO detects the merger of two neutron stars

On the Quantum Footpath

The concept of the trajectory of a quantum particle almost vanished in the battle between Werner Heisenberg’s matrix mechanics and Erwin Schrödinger’s wave mechanics.  It took Niels Bohr and his complementarity principle of wave-particle duality to cede back some reality to quantum trajectories.  However, Schrödinger and Einstein were not convinced and conceived of quantum entanglement to refute the growing acceptance of the Copenhagen Interpretation of quantum physics.  Schrödinger’s cat was meant to be an absurdity, but ironically it has become a central paradigm of practical quantum computers.  Quantum trajectories took on new meaning when Richard Feynman constructed quantum theory based on the principle of least action, inventing his famous Feynman Diagrams to help explain quantum electrodynamics.

1885    Balmer Theory: 

1897    J. J. Thomson discovered the electron

1904    Thomson plum pudding model of the atom

1911    Bohr PhD thesis filed. Studies on the electron theory of metals.  Visited England.

1911    Rutherford nuclear model

1911    First Solvay conference

1911    “ultraviolet catastrophe” coined by Ehrenfest

1913    Bohr combined Rutherford’s nuclear atom with Planck’s quantum hypothesis: 1913 Bohr model

1913    Ehrenfest adiabatic hypothesis

1914-1916       Bohr at Manchester with Rutherford

1916    Bohr appointed Chair of Theoretical Physics at University of Copenhagen: a position that was made just for him

1916    Schwarzschild and Epstein introduce action-angle coordinates into quantum theory

1920    Heisenberg enters University of Munich to obtain his doctorate

1920    Bohr’s Correspondence principle: Classical physics for large quantum numbers

1921    Bohr Founded Institute of Theoretical Physics (Copenhagen)

1922-1923       Heisenberg studies with Born, Franck and Hilbert at Göttingen while Sommerfeld is in the US on sabbatical.

1923    Heisenberg Doctorate.  The exam does not go well.  Unable to derive the resolving power of a microscope in response to question by Wien.  Becomes Born’s assistant at Göttingen.

1924    Heisenberg visits Niels Bohr in Copenhagen (and met Einstein?)

1924    Heisenberg Habilitation at Göttingen on anomalous Zeeman

1924 – 1925    Heisenberg worked with Bohr in Copenhagen, returned summer of 1925 to Göttiingen

1924    Pauli exclusion principle and state occupancy

1924    de Broglie hypothesis extended wave-particle duality to matter

1924    Bohr Predicted Halfnium (72)

1924    Kronig’s proposal for electron self spin

1924    Bose (Einstein)

1925    Heisenberg paper on quantum mechanics

1925    Dirac, reading proof from Heisenberg, recognized the analogy of noncommutativity with Poisson brackets and the correspondence with Hamiltonian mechanics.

1925    Uhlenbeck and Goudschmidt: spin

1926    Born, Heisenberg, Kramers: virtual oscillators at transition frequencies: Matrix mechanics (alternative to Bohr-Kramers-Slater 1924 model of orbits).  Heisenberg was Born’s student at Göttingen.

1926    Schrödinger wave mechanics

1927    de Broglie hypotehsis confirmed by Davisson and Germer

1927    Complementarity by Bohr: wave-particle duality “Evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.

1927    Heisenberg uncertainty principle (Heisenberg was in Copenhagen 1926 – 1927)

1927    Solvay Conference in Brussels

1928    Heisenberg to University of Leipzig

1928    Dirac relativistic QM equation

1929    de Broglie Nobel Prize

1930    Solvay Conference

1932    Heisenberg Nobel Prize

1932    von Neumann operator algebra

1933    Dirac Lagrangian form of QM (basis of Feynman path integral)

1933    Schrödinger and Dirac Nobel Prize

1935    Einstein, Poldolsky and Rosen EPR paper

1935 Bohr’s response to Einsteins “EPR” paradox

1935    Schrodinger’s cat

1939    Feynman graduates from MIT

1941    Heisenberg (head of German atomic project) visits Bohr in Copenhagen

1942    Feynman PhD at Princeton, “The Principle of Least Action in Quantum Mechanics

1942 – 1945    Manhattan Project, Bethe-Feynman equation for fission yield

1943    Bohr escapes to Sweden in a fishing boat.  Went on to England secretly.

1945    Pauli Nobel Prize

1945    Death of Feynman’s wife Arline (married 4 years)

1945    Fall, Feynman arrives at Cornell ahead of Hans Bethe

1947    Shelter Island conference: Lamb Shift, did Kramer’s give a talk suggesting that infinities could be subtracted?

1947    Fall, Dyson arrives at Cornell

1948    Pocono Manor, Pennsylvania, troubled unveiling of path integral formulation and Feynman diagrams, Schwinger’s master presentation

1948    Feynman and Dirac. Summer drive across the US with Dyson

1949    Dyson joins IAS as a postdoc, trains a cohort of theorists in Feynman’s technique

1949    Karplus and Kroll first g-factor calculation

1950    Feynman moves to Cal Tech

1965    Schwinger, Tomonaga and Feynman Nobel Prize

1967    Hans Bethe Nobel Prize

From Butterflies to Hurricanes

Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable.  In the hands of Vladimir Arnold and Jürgen Moser, this became the KAM theory of Hamiltonian chaos.  This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory.  Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor.  Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.

1760    Euler 3-body problem (two fixed centers and coplanar third body)

1763    Euler colinear 3-body problem

1772    Lagrange equilateral 3-body problem

1881-1886       Poincare memoires “Sur les courbes de ́finies par une equation differentielle”

1890    Poincare “Sur le probleme des trois corps et les equations de la dynamique”. First-return map, Poincare recurrence theorem, stable and unstable manifolds

1892 – 1899    Poincare New Methods in Celestial Mechanics

1892    Lyapunov The General Problem of the Stability of Motion

1899    Poincare homoclinic trajectory

1913    Birkhoff proves Poincaré’s last geometric theorem, a special case of the three-body problem.

1927    van der Pol and van der Mark

1937    Coarse systems, Andronov and Pontryagin

1938    Morse theory

1942    Hopf bifurcation

1945    Cartwright and Littlewood study the van der Pol equation (Radar during WWII)

1954    Kolmogorov A. N., On conservation of conditionally periodic motions for a small change in Hamilton’s function.

1960    Lorenz: 12 equations

1962    Moser On Invariant Curves of Area-Preserving Mappings of an Annulus.

1963    Arnold Small denominators and problems of the stability of motion in classical and celestial mechanics

1963    Lorenz: 3 equations

1964    Arnold diffusion

1965    Smale’s horseshoe

1969    Chirikov standard map

1971    Ruelle-Takens (Ruelle coins phrase “strange attractor”)

1972    “Butterfly Effect” given for Lorenz’ talk (by Philip Merilees)

1975    Gollub-Swinney observe route to turbulence along lines of Ruelle

1975    Yorke coins “chaos theory”

1976    Robert May writes review article of the logistic map

1977    New York conference on bifurcation theory

1987    James Gleick Chaos: Making a New Science

Darwin in the Clockworks

The preceding timelines related to the central role played by families of trajectories phase space to explain the time evolution of complex systems.  These ideas are extended to explore the history and development of the theory of natural evolution by Charles Darwin.  Darwin had many influences, including ideas from Thomas Malthus in the context of economic dynamics.  After Darwin, the ideas of evolution matured to encompass broad topics in evolutionary dynamics and the emergence of the idea of fitness landscapes and game theory driving the origin of new species.  The rise of genetics with Gregor Mendel supplied a firm foundation for molecular evolution, leading to the moleculer clock of Linus Pauling and the replicator dynamics of Richard Dawkins.

1202    Fibonacci

1766    Thomas Robert Malthus born

1776    Adam Smith The Wealth of Nations

1798    Malthus “An Essay on the Principle of Population

1817    Ricardo Principles of Political Economy and Taxation

1838    Cournot early equilibrium theory in duopoly

1848    John Stuart Mill

1848    Karl Marx Communist Manifesto

1859    Darwin Origin of Species

1867    Karl Marx Das Kapital

1871    Darwin Descent of Man, and Selection in Relation to Sex

1871    Jevons Theory of Political Economy

1871    Menger Principles of Economics

1874    Walrus Éléments d’économie politique pure, or Elements of Pure Economics (1954)

1890    Marshall Principles of Economics

1908    Hardy constant genetic variance

1910    Brouwer fixed point theorem

1910    Alfred J. Lotka autocatylitic chemical reactions

1913    Zermelo determinancy in chess

1922    Fisher dominance ratio

1922    Fisher mutations

1925    Lotka predator-prey in biomathematics

1926    Vita Volterra published same equations independently

1927    JBS Haldane (1892—1964) mutations

1928    von Neumann proves the minimax theorem

1930    Fisher ratio of sexes

1932    Wright Adaptive Landscape

1932    Haldane The Causes of Evolution

1933    Kolmogorov Foundations of the Theory of Probability

1934    Rudolph Carnap The Logical Syntax of Language

1936    John Maynard Keynes, The General Theory of Employment, Interest and Money

1936    Kolmogorov generalized predator-prey systems

1938    Borel symmetric payoff matrix

1942    Sewall Wright    Statistical Genetics and Evolution

1943    McCulloch and Pitts A Logical Calculus of Ideas Immanent in Nervous Activity

1944    von Neumann and Morgenstern Theory of Games and Economic Behavior

1950    Prisoner’s Dilemma simulated at Rand Corportation

1950    John Nash Equilibrium points in n-person games and The Bargaining Problem

1951    John Nash Non-cooperative Games

1952    McKinsey Introduction to the Theory of Games (first textbook)

1953    John Nash Two-Person Cooperative Games

1953    Watson and Crick DNA

1955    Braithwaite’s Theory of Games as a Tool for the Moral Philosopher

1961    Lewontin Evolution and the Theory of Games

1962    Patrick Moran The Statistical Processes of Evolutionary Theory

1962    Linus Pauling molecular clock

1968    Motoo Kimura  neutral theory of molecular evolution

1972    Maynard Smith introduces the evolutionary stable solution (ESS)

1972    Gould and Eldridge Punctuated equilibrium

1973    Maynard Smith and Price The Logic of Animal Conflict

1973    Black Scholes

1977    Eigen and Schuster The Hypercycle

1978    Replicator equation (Taylor and Jonker)

1982    Hopfield network

1982    John Maynard Smith Evolution and the Theory of Games

1984    R. Axelrod The Evolution of Cooperation

The Measure of Life

This final topic extends the ideas of dynamics into abstract spaces of high dimension to encompass the idea of a trajectory of life.  Health and disease become dynamical systems defined by all the proteins and nucleic acids that comprise the physical self.  Concepts from network theory, autonomous oscillators and synchronization contribute to this viewpoint.  Healthy trajectories are like stable limit cycles in phase space, but disease can knock the system trajectory into dangerous regions of health space, as doctors turn to new developments in personalized medicine try to return the individual to a healthy path.  This is the ultimate generalization of Galileo’s simple parabolic trajectory.

1642    Galileo dies

1656    Huygens invents pendulum clock

1665    Huygens observes “odd kind of sympathy” in synchronized clocks

1673    Huygens publishes Horologium Oscillatorium sive de motu pendulorum

1736    Euler Seven Bridges of Königsberg

1845    Kirchhoff’s circuit laws

1852    Guthrie four color problem

1857    Cayley trees

1858    Hamiltonian cycles

1887    Cajal neural staining microscopy

1913    Michaelis Menten dynamics of enzymes

1924    Berger, Hans: neural oscillations (Berger invented the EEG)

1926    van der Pol dimensioness form of equation

1927    van der Pol periodic forcing

1943    McCulloch and Pits mathematical model of neural nets

1948    Wiener cybernetics

1952    Hodgkin and Huxley action potential model

1952    Turing instability model

1956    Sutherland cyclic AMP

1957    Broadbent and Hammersley bond percolation

1958    Rosenblatt perceptron

1959    Erdös and Renyi random graphs

1962    Cohen EGF discovered

1965    Sebeok coined zoosemiotics

1966    Mesarovich systems biology

1967    Winfree biological rythms and coupled oscillators

1969    Glass Moire patterns in perception

1970    Rodbell G-protein

1971    phrase “strange attractor” coined (Ruelle)

1972    phrase “signal transduction” coined (Rensing)

1975    phrase “chaos theory” coined (Yorke)

1975    Werbos backpropagation

1975    Kuramoto transition

1976    Robert May logistic map

1977    Mackey-Glass equation and dynamical disease

1982    Hopfield network

1990    Strogatz and Murillo pulse-coupled oscillators

1997    Tomita systems biology of a cell

1998    Strogatz and Watts Small World network

1999    Barabasi Scale Free networks

2000    Sequencing of the human genome

A Commotion in the Stars: The Legacy of Christian Doppler

Christian Andreas Doppler (1803 – 1853) was born in Salzburg, Austria, to a longstanding family of stonemasons.  As a second son, he was expected to help his older brother run the business, so his Father had him tested in his 18th year for his suitability for a career in business.  The examiner Simon Stampfer (1790 – 1864), an Austrian mathematician and inventor teaching at the Lyceum in Salzburg, discovered that Doppler had a gift for mathematics and was better suited for a scientific career.  Stampfer’s enthusiasm convinced Doppler’s father to enroll him in the Polytechnik Institute in Vienna (founded only a few years earlier in 1815) where he took classes in mathematics, mechanics and physics [1] from 1822 to 1825.  Doppler excelled in his courses, but was dissatisfied with the narrowness of the education, yearning for more breadth and depth in his studies and for more significance in his positions, feelings he would struggle with for his entire short life.  He left Vienna, returning to the Lyceum in Salzburg to round out his education with philosophy, languages and poetry.  Unfortunately, this four-year detour away from technical studies impeded his ability to gain a permanent technical position, so he began a temporary assistantship with a mathematics professor at Vienna.  As he approached his 30th birthday this term expired without prospects.  He was about to emigrate to America when he finally received an offer to teach at a secondary school in Prague.

To read about the attack by Joseph Petzval on Doppler’s effect and the effect it had on Doppler, see my feature article “The Fall and Rise of the Doppler Effect in Physics Today, 73(3) 30, March (2020).

Salzburg Austria

Doppler in Prague

Prague gave Doppler new life.  He was a professor with a position that allowed him to marry the daughter of a sliver and goldsmith from Salzburg.  He began to publish scholarly papers, and in 1837 was appointed supplementary professor of Higher Mathematics and Geometry at the Prague Technical Institute, promoted to full professor in 1841.  It was here that he met the unusual genius Bernard Bolzano (1781 – 1848), recently returned from political exile in the countryside.  Bolzano was a philosopher and mathematician who developed rigorous concepts of mathematical limits and is famous today for his part in the Bolzano-Weierstrass theorem in functional analysis, but he had been too liberal and too outspoken for the conservative Austrian regime and had been dismissed from the University in Prague in 1819.  He was forbidden to publish his work in Austrian journals, which is one reason why much of Bolzano’s groundbreaking work in functional analysis remained unknown during his lifetime.  However, he participated in the Bohemian Society for Science from a distance, recognizing the inventive tendencies in the newcomer Doppler and supporting him for membership in the Bohemian Society.  When Bolzano was allowed to return in 1842 to the Polytechnic Institute in Prague, he and Doppler became close friends as kindred spirits. 

Prague, Czech Republic

On May 25, 1842, Bolzano presided as chairman over a meeting of the Bohemian Society for Science on the day that Doppler read a landmark paper on the color of stars to a meagre assembly of only five regular members of the Society [2].  The turn-out was so small that the meeting may have been held in the robing room of the Society rather than in the meeting hall itself.  Leading up to this famous moment, Doppler’s interests were peripatetic, ranging widely over mathematical and physical topics, but he had lately become fascinated by astronomy and by the phenomenon of stellar aberration.  Stellar aberration was discovered by James Bradley in 1729 and explained as the result of the Earth’s yearly motion around the Sun, causing the apparent location of a distant star to change slightly depending on the direction of the Earth’s motion.  Bradley explained this in terms of the finite speed of light and was able to estimate it to within several percent [3].  As Doppler studied Bradley aberration, he wondered how the relative motion of the Earth would affect the color of the star.  By making a simple analogy of a ship traveling with, or against, a series of ocean waves, he concluded that the frequency of impact of the peaks and troughs of waves on the ship was no different than the arrival of peaks and troughs of the light waves impinging on the eye.  Because perceived color was related to the frequency of excitation in the eye, he concluded that the color of light would be slightly shifted to the blue if approaching, and to the red if receding from, the light source. 

Doppler wave fronts from a source emitting spherical waves moving with speeds β relative to the speed of the wave in the medium.

Doppler calculated the magnitude of the effect by taking a simple ratio of the speed of the observer relative to the speed of light.  What he found was that the speed of the Earth, though sufficient to cause the detectable aberration in the position of stars, was insufficient to produce a noticeable change in color.  However, his interest in astronomy had made him familiar with binary stars where the relative motion of the light source might be high enough to cause color shifts.  In fact, in the star catalogs there were examples of binary stars that had complementary red and blue colors.  Therefore, the title of his paper, published in the Proceedings of the Royal Bohemian Society of Sciences a few months after he read it to the society, was “On the Coloured Light of the Double Stars and Certain Other Stars of the Heavens: Attempt at a General Theory which Incorporates Bradley’s Theorem of Aberration as an Integral Part” [4]

Title page of Doppler’s 1842 paper introducing the Doppler Effect.

Doppler’s analogy was correct, but like all analogies not founded on physical law, it differed in detail from the true nature of the phenomenon.  By 1842 the transverse character of light waves had been thoroughly proven through the work of Fresnel and Arago several decades earlier, yet Doppler held onto the old-fashioned notion that light was composed of longitudinal waves.  Bolzano, fully versed in the transverse nature of light, kindly published a commentary shortly afterwards [5] showing how the transverse effect for light, and a longitudinal effect for sound, were both supported by Doppler’s idea.  Yet Doppler also did not know that speeds in visual binaries were too small to produce noticeable color effects to the unaided eye.  Finally, (and perhaps the greatest flaw in his argument on the color of stars) a continuous spectrum that extends from the visible into the infrared and ultraviolet would not change color because all the frequencies would shift together preserving the flat (white) spectrum.

The simple algebraic derivation of the Doppler Effect in the 1842 publication..

Doppler’s twelve years in Prague were intense.  He was consumed by his Society responsibilities and by an extremely heavy teaching load that included personal exams of hundreds of students.  The only time he could be creative was during the night while his wife and children slept.  Overworked and running on too little rest, his health already frail with the onset of tuberculosis, Doppler collapsed, and he was unable to continue at the Polytechnic.  In 1847 he transferred to the School of Mines and Forrestry in Schemnitz (modern Banská Štiavnica in Slovakia) with more pay and less work.  Yet the revolutions of 1848 swept across Europe, with student uprisings, barricades in the streets, and Hungarian liberation armies occupying the cities and universities, giving him no peace.  Providentially, his former mentor Stampfer retired from the Polytechnic in Vienna, and Doppler was called to fill the vacancy.

Although Doppler was named the Director of Austria’s first Institute of Physics and was elected to the National Academy, he ran afoul of one of the other Academy members, Joseph Petzval (1807 – 1891), who persecuted Doppler and his effect.  To read a detailed description of the attack by Petzval on Doppler’s effect and the effect it had on Doppler, see my feature article “The Fall and Rise of the Doppler Effect” in Physics Today, March issue (2020).

Christian Doppler

Voigt’s Transformation

It is difficult today to appreciate just how deeply engrained the reality of the luminiferous ether was in the psyche of the 19th century physicist.  The last of the classical physicists were reluctant even to adopt Maxwell’s electromagnetic theory for the explanation of optical phenomena, and as physicists inevitably were compelled to do so, some of their colleagues looked on with dismay and disappointment.  This was the situation for Woldemar Voigt (1850 – 1919) at the University of Göttingen, who was appointed as one of the first professors of physics there in 1883, to be succeeded in later years by Peter Debye and Max Born.  Voigt received his doctorate at the University of Königsberg under Franz Neumann, exploring the elastic properties of rock salt, and at Göttingen he spent a quarter century pursuing experimental and theoretical research into crystalline properties.  Voigt’s research, with students like Paul Drude, laid the foundation for the modern field of solid state physics.  His textbook Lehrbuch der Kristallphysik published in 1910 remained influential well into the 20th century because it adopted mathematical symmetry as a guiding principle of physics.  It was in the context of his studies of crystal elasticity that he introduced the word “tensor” into the language of physics.

At the January 1887 meeting of the Royal Society of Science at Göttingen, three months before Michelson and Morely began their reality-altering experiments at the Case Western Reserve University in Cleveland Ohio, Voit submitted a paper deriving the longitudinal optical Doppler effect in an incompressible medium.  He was responding to results published in 1886 by Michelson and Morely on their measurements of the Fresnel drag coefficient, which was the precursor to their later results on the absolute motion of the Earth through the ether. 

Fresnel drag is the effect of light propagating through a medium that is in motion.  The French physicist Francois Arago (1786 – 1853) in 1810 had attempted to observe the effects of corpuscles of light emitted from stars propagating with different speeds through the ether as the Earth spun on its axis and traveled around the sun.  He succeeded only in observing ordinary stellar aberration.  The absence of the effects of motion through the ether motivated Augustin-Jean Fresnel (1788 – 1827) to apply his newly-developed wave theory of light to explain the null results.  In 1818 Fresnel derived an expression for the dragging of light by a moving medium that explained the absence of effects in Arago’s observations.  For light propagating through a medium of refractive index n that is moving at a speed v, the resultant velocity of light is

where the last term in parenthesis is the Fresnel drag coefficient.  The Fresnel drag effect supported the idea of the ether by explaining why its effects could not be observed—a kind of Catch-22—but it also applied to light moving through a moving dielectric medium.  In 1851, Fizeau used an interferometer to measure the Fresnel drag coefficient for light moving through moving water, arriving at conclusions that directly confirmed the Fresnel drag effect.  The positive experiments of Fizeau, as well as the phenomenon of stellar aberration, would be extremely influential on the thoughts of Einstein as he developed his approach to special relativity in 1905.  They were also extremely influential to Michelson, Morley and Voigt.

 In his paper on the absence of the Fresnel drag effect in the first Michelson-Morley experiment, Voigt pointed out that an equation of the form

is invariant under the transformation

From our modern vantage point, we immediately recognize (to within a scale factor) the Lorentz transformation of relativity theory.  The first equation is common Galilean relativity, but the last equation was something new, introducing a position-dependent time as an observer moved with speed  relative to the speed of light [6].  Using these equations, Voigt was the first to derive the longitudinal (conventional) Doppler effect from relativistic effects.

Voigt’s derivation of the longitudinal Doppler effect used a classical approach that is still used today in Modern Physics textbooks to derive the Doppler effect.  The argument proceeds by considering a moving source that emits a continuous wave in the direction of motion.  Because the wave propagates at a finite speed, the moving source chases the leading edge of the wave front, catching up by a small amount by the time a single cycle of the wave has been emitted.  The resulting compressed oscillation represents a blue shift of the emitted light.  By using his transformations, Voigt arrived at the first relativistic expression for the shift in light frequency.  At low speeds, Voigt’s derivation reverted to Doppler’s original expression.

A few months after Voigt delivered his paper, Michelson and Morley announced the results of their interferometric measurements of the motion of the Earth through the ether—with their null results.  In retrospect, the Michelson-Morley experiment is viewed as one of the monumental assaults on the old classical physics, helping to launch the relativity revolution.  However, in its own day, it was little more than just another null result on the ether.  It did incite Fitzgerald and Lorentz to suggest that length of the arms of the interferometer contracted in the direction of motion, with the eventual emergence of the full Lorentz transformations by 1904—seventeen years after the Michelson results.

            In 1904 Einstein, working in relative isolation at the Swiss patent office, was surprisingly unaware of the latest advances in the physics of the ether.  He did not know about Voigt’s derivation of the relativistic Doppler effect  (1887) as he had not heard of Lorentz’s final version of relativistic coordinate transformations (1904).  His thinking about relativistic effects focused much farther into the past, to Bradley’s stellar aberration (1725) and Fizeau’s experiment of light propagating through moving water (1851).  Einstein proceeded on simple principles, unencumbered by the mental baggage of the day, and delivered his beautifully minimalist theory of special relativity in his famous paper of 1905 “On the Electrodynamics of Moving Bodies”, independently deriving the Lorentz coordinate transformations [7]

One of Einstein’s talents in theoretical physics was to predict new phenomena as a way to provide direct confirmation of a new theory.  This was how he later famously predicted the deflection of light by the Sun and the gravitational frequency shift of light.  In 1905 he used his new theory of special relativity to predict observable consequences that included a general treatment of the relativistic Doppler effect.  This included the effects of time dilation in addition to the longitudinal effect of the source chasing the wave.  Time dilation produced a correction to Doppler’s original expression for the longitudinal effect that became significant at speeds approaching the speed of light.  More significantly, it predicted a transverse Doppler effect for a source moving along a line perpendicular to the line of sight to an observer.  This effect had not been predicted either by Doppler or by Voigt.  The equation for the general Doppler effect for any observation angle is

Just as Doppler had been motivated by Bradley’s aberration of starlight when he conceived of his original principle for the longitudinal Doppler effect, Einstein combined the general Doppler effect with his results for the relativistic addition of velocities (also in his 1905 Annalen paper) as the conclusive treatment of stellar aberration nearly 200 years after Bradley first observed the effect.

Despite the generally positive reception of Einstein’s theory of special relativity, some of its consequences were anathema to many physicists at the time.  A key stumbling block was the question whether relativistic effects, like moving clocks running slowly, were only apparent, or were actually real, and Einstein had to fight to convince others of its reality.  When Johannes Stark (1874 – 1957) observed Doppler line shifts in ion beams called “canal rays” in 1906 (Stark received the 1919 Nobel prize in part for this discovery) [8], Einstein promptly published a paper suggesting how the canal rays could be used in a transverse geometry to directly detect time dilation through the transverse Doppler effect [9].  Thirty years passed before the experiment was performed with sufficient accuracy by Herbert Ives and G. R. Stilwell in 1938 to measure the transverse Doppler effect [10].  Ironically, even at this late date, Ives and Stilwell were convinced that their experiment had disproved Einstein’s time dilation by supporting Lorentz’ contraction theory of the electron.  The Ives-Stilwell experiment was the first direct test of time dilation, followed in 1940 by muon lifetime measurements [11].

Further Reading

D. D. Nolte, “The Fall and Rise of the Doppler Effect“, Phys. Today 73(3) 30, March 2020.


[1] pg. 15, Eden, A. (1992). The search for Christian Doppler. Wien, Springer-Verlag.

[2] pg. 30, Eden

[3] Bradley, J (1729). “Account of a new discoved Motion of the Fix’d Stars”. Phil Trans. 35: 637–660.

[4] C. A. DOPPLER, “Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (About the coloured light of the binary stars and some other stars of the heavens),” Proceedings of the Royal Bohemian Society of Sciences, vol. V, no. 2, pp. 465–482, (Reissued 1903) (1842).

[5] B. Bolzano, “Ein Paar Bemerkunen über die Neu Theorie in Herrn Professor Ch. Doppler’s Schrift “Über das farbige Licht der Doppersterne und eineger anderer Gestirnedes Himmels”,” Pogg. Anal. der Physik und Chemie, vol. 60, p. 83, 1843; B. Bolzano, “Christian Doppler’s neuste Leistunen af dem Gebiet der physikalischen Apparatenlehre, Akoustik, Optik and optische Astronomie,” Pogg. Anal. der Physik und Chemie, vol. 72, pp. 530-555, 1847.

[6] W. Voigt, “Uber das Doppler’sche Princip,” Göttinger Nachrichten, vol. 7, pp. 41–51, (1887). The common use of c to express the speed of light came later from Voigt’s student Paul Drude.

[7] A. Einstein, “On the electrodynamics of moving bodies,” Annalen Der Physik, vol. 17, pp. 891-921, 1905.

[8] J. Stark, W. Hermann, and S. Kinoshita, “The Doppler effect in the spectrum of mercury,” Annalen Der Physik, vol. 21, pp. 462-469, Nov 1906.

[9] A. Einstein, “”Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips”,” vol. 328, pp. 197–198, 1907.

[10] H. E. Ives and G. R. Stilwell, “An experimental study of the rate of a moving atomic clock,” Journal of the Optical Society of America, vol. 28, p. 215, 1938.

[11] B. Rossi and D. B. Hall, “Variation of the Rate of Decay of Mesotrons with Momentum,” Physical Review, vol. 59, pp. 223–228, 1941.

Bohr’s Orbits

The first time I ran across the Bohr-Sommerfeld quantization conditions I admit that I laughed! I was a TA for the Modern Physics course as a graduate student at Berkeley in 1982 and I read about Bohr-Sommerfeld in our Tipler textbook. I was familiar with Bohr orbits, which are already the wrong way of thinking about quantized systems. So the Bohr-Sommerfeld conditions, especially for so-called “elliptical” orbits, seemed like nonsense.

But it’s funny how a a little distance gives you perspective. Forty years later I know a little more physics than I did then, and I have gained a deep respect for an obscure property of dynamical systems known as “adiabatic invariants”. It turns out that adiabatic invariants lie at the core of quantum systems, and in the case of hydrogen adiabatic invariants can be visualized as … elliptical orbits!

Quantum Physics in Copenhagen

Niels Bohr (1885 – 1962) was born in Copenhagen, Denmark, the middle child of a physiology professor at the University in Copenhagen.  Bohr grew up with his siblings as a faculty child, which meant an unconventional upbringing full of ideas, books and deep discussions.  Bohr was a late bloomer in secondary school but began to show talent in Math and Physics in his last two years.  When he entered the University in Copenhagen in 1903 to major in physics, the university had only one physics professor, Christian Christiansen, and had no physics laboratories.  So Bohr tinkered in his father’s physiology laboratory, performing a detailed experimental study of the hydrodynamics of water jets, writing and submitting a paper that was to be his only experimental work.  Bohr went on to receive a Master’s degree in 1909 and his PhD in 1911, writing his thesis on the theory of electrons in metals.  Although the thesis did not break much new ground, it uncovered striking disparities between observed properties and theoretical predictions based on the classical theory of the electron.  For his postdoc studies he applied for and was accepted to a position working with the discoverer of the electron, Sir J. J. Thompson, in Cambridge.  Perhaps fortunately for the future history of physics, he did not get along well with Thompson, and he shifted his postdoc position in early 1912 to work with Ernest Rutherford at the much less prestigious University of Manchester.

Niels Bohr (Wikipedia)

Ernest Rutherford had just completed a series of detailed experiments on the scattering of alpha particles on gold film and had demonstrated that the mass of the atom was concentrated in a very small volume that Rutherford called the nucleus, which also carried the positive charge compensating the negative electron charges.  The discovery of the nucleus created a radical new model of the atom in which electrons executed planetary-like orbits around the nucleus.  Bohr immediately went to work on a theory for the new model of the atom.  He worked closely with Rutherford and the other members of Rutherford’s laboratory, involved in daily discussions on the nature of atomic structure.  The open intellectual atmosphere of Rutherford’s group and the ready flow of ideas in group discussions became the model for Bohr, who would some years later set up his own research center that would attract the top young physicists of the time.  Already by mid 1912, Bohr was beginning to see a path forward, hinting in letters to his younger brother Harald (who would become a famous mathematician) that he had uncovered a new approach that might explain some of the observed properties of simple atoms. 

By the end of 1912 his postdoc travel stipend was over, and he returned to Copenhagen, where he completed his work on the hydrogen atom.  One of the key discrepancies in the classical theory of the electron in atoms was the requirement, by Maxwell’s Laws, for orbiting electrons to continually radiate because of their angular acceleration.  Furthermore, from energy conservation, if they radiated continuously, the electron orbits must also eventually decay into the nuclear core with ever-decreasing orbital periods and hence ever higher emitted light frequencies.  Experimentally, on the other hand, it was known that light emitted from atoms had only distinct quantized frequencies.  To circumvent the problem of classical radiation, Bohr simply assumed what was observed, formulating the idea of stationary quantum states.  Light emission (or absorption) could take place only when the energy of an electron changed discontinuously as it jumped from one stationary state to another, and there was a lowest stationary state below which the electron could never fall.  He then took a critical and important step, combining this new idea of stationary states with Planck’s constant h.  He was able to show that the emission spectrum of hydrogen, and hence the energies of the stationary states, could be derived if the angular momentum of the electron in a Hydrogen atom was quantized by integer amounts of Planck’s constant h

Bohr published his quantum theory of the hydrogen atom in 1913, which immediately focused the attention of a growing group of physicists (including Einstein, Rutherford, Hilbert, Born, and Sommerfeld) on the new possibilities opened up by Bohr’s quantum theory [1].  Emboldened by his growing reputation, Bohr petitioned the university in Copenhagen to create a new faculty position in theoretical physics, and to appoint him to it.  The University was not unreceptive, but university bureaucracies make decisions slowly, so Bohr returned to Rutherford’s group in Manchester while he awaited Copenhagen’s decision.  He waited over two years, but he enjoyed his time in the stimulating environment of Rutherford’s group in Manchester, growing steadily into the role as master of the new quantum theory.  In June of 1916, Bohr returned to Copenhagen and a year later was elected to the Royal Danish Academy of Sciences. 

Although Bohr’s theory had succeeded in describing some of the properties of the electron in atoms, two central features of his theory continued to cause difficulty.  The first was the limitation of the theory to single electrons in circular orbits, and the second was the cause of the discontinuous jumps.  In response to this challenge, Arnold Sommerfeld provided a deeper mechanical perspective on the origins of the discrete energy levels of the atom. 

Quantum Physics in Munich

Arnold Johannes Wilhem Sommerfeld (1868—1951) was born in Königsberg, Prussia, and spent all the years of his education there to his doctorate that he received in 1891.  In Königsberg he was acquainted with Minkowski, Wien and Hilbert, and he was the doctoral student of Lindemann.  He also was associated with a social group at the University that spent too much time drinking and dueling, a distraction that lead to his receiving a deep sabre cut on his forehead that became one of his distinguishing features along with his finely waxed moustache.  In outward appearance, he looked the part of a Prussian hussar, but he finally escaped this life of dissipation and landed in Göttingen where he became Felix Klein’s assistant in 1894.  He taught at local secondary schools, rising in reputation, until he secured a faculty position of theoretical physics at the University in Münich in 1906.  One of his first students was Peter Debye who received his doctorate under Sommerfeld in 1908.  Later famous students would include Peter Ewald (doctorate in 1912), Wolfgang Pauli (doctorate in 1921), Werner Heisenberg (doctorate in 1923), and Hans Bethe (doctorate in 1928).  These students had the rare treat, during their time studying under Sommerfeld, of spending weekends in the winter skiing and staying at a ski hut that he owned only two hours by train outside of Münich.  At the end of the day skiing, discussion would turn invariably to theoretical physics and the leading problems of the day.  It was in his early days at Münich that Sommerfeld played a key role aiding the general acceptance of Minkowski’s theory of four-dimensional space-time by publishing a review article in Annalen der Physik that translated Minkowski’s ideas into language that was more familiar to physicists.

Arnold Sommerfeld (Wikipedia)

Around 1911, Sommerfeld shifted his research interest to the new quantum theory, and his interest only intensified after the publication of Bohr’s model of hydrogen in 1913.  In 1915 Sommerfeld significantly extended the Bohr model by building on an idea put forward by Planck.  While further justifying the black body spectrum, Planck turned to descriptions of the trajectory of a quantized one-dimensional harmonic oscillator in phase space.  Planck had noted that the phase-space areas enclosed by the quantized trajectories were integral multiples of his constant.  Sommerfeld expanded on this idea, showing that it was not the area enclosed by the trajectories that was fundamental, but the integral of the momentum over the spatial coordinate [2].  This integral is none other than the original action integral of Maupertuis and Euler, used so famously in their Principle of Least Action almost 200 years earlier.  Where Planck, in his original paper of 1901, had recognized the units of his constant to be those of action, and hence called it the quantum of action, Sommerfeld made the explicit connection to the dynamical trajectories of the oscillators.  He then showed that the same action principle applied to Bohr’s circular orbits for the electron on the hydrogen atom, and that the orbits need not even be circular, but could be elliptical Keplerian orbits. 

The quantum condition for this otherwise classical trajectory was the requirement for the action integral over the motion to be equal to integer units of the quantum of action.  Furthermore, Sommerfeld showed that there must be as many action integrals as degrees of freedom for the dynamical system.  In the case of Keplerian orbits, there are radial coordinates as well as angular coordinates, and each action integral was quantized for the discrete electron orbits.  Although Sommerfeld’s action integrals extended Bohr’s theory of quantized electron orbits, the new quantum conditions also created a problem because there were now many possible elliptical orbits that all had the same energy.  How was one to find the “correct” orbit for a given orbital energy?

Quantum Physics in Leiden

In 1906, the Austrian Physicist Paul Ehrenfest (1880 – 1933), freshly out of his PhD under the supervision of Boltzmann, arrived at Göttingen only weeks before Boltzmann took his own life.  Felix Klein at Göttingen had been relying on Boltzmann to provide a comprehensive review of statistical mechanics for the Mathematical Encyclopedia, so he now entrusted this project to the young Ehrenfest.  It was a monumental task, which was to take him and his physicist wife Tatyana nearly five years to complete.  Part of the delay was the desire by Ehrenfest to close some open problems that remained in Boltzmann’s work.  One of these was a mechanical theorem of Boltzmann’s that identified properties of statistical mechanical systems that remained unaltered through a very slow change in system parameters.  These properties would later be called adiabatic invariants by Einstein.  Ehrenfest recognized that Wien’s displacement law, which had been a guiding light for Planck and his theory of black body radiation, had originally been derived by Wien using classical principles related to slow changes in the volume of a cavity.  Ehrenfest was struck by the fact that such slow changes would not induce changes in the quantum numbers of the quantized states, and hence that the quantum numbers must be adiabatic invariants of the black body system.  This not only explained why Wien’s displacement law continued to hold under quantum as well as classical considerations, but it also explained why Planck’s quantization of the energy of his simple oscillators was the only possible choice.  For a classical harmonic oscillator, the ratio of the energy of oscillation to the frequency of oscillation is an adiabatic invariant, which is immediately recognized as Planck’s quantum condition .  

Paul Ehrenfest (Wikipedia)

Ehrenfest published his observations in 1913 [3], the same year that Bohr published his theory of the hydrogen atom, so Ehrenfest immediately applied the theory of adiabatic invariants to Bohr’s model and discovered that the quantum condition for the quantized energy levels was again the adiabatic invariants of the electron orbits, and not merely a consequence of integer multiples of angular momentum, which had seemed somewhat ad hoc.  Later, when Sommerfeld published his quantized elliptical orbits in 1916, the multiplicity of quantum conditions and orbits had caused concern, but Ehrenfest came to the rescue with his theory of adiabatic invariants, showing that each of Sommerfeld’s quantum conditions were precisely the adabatic invariants of the classical electron dynamics [4]. The remaining question was which coordinates were the correct ones, because different choices led to different answers.  This was quickly solved by Johannes Burgers (one of Ehrenfest’s students) who showed that action integrals were adiabatic invariants, and then by Karl Schwarzschild and Paul Epstein who showed that action-angle coordinates were the only allowed choice of coordinates, because they enabled the separation of the Hamilton-Jacobi equations and hence provided the correct quantization conditions for the electron orbits.  Schwarzshild’s paper was published the same day that he died on the Eastern Front.  The work by Schwarzschild and Epstein was the first to show the power of the Hamiltonian formulation of dynamics for quantum systems, which foreshadowed the future importance of Hamiltonians for quantum theory.

Karl Schwarzschild (Wikipedia)


Emboldened by Ehrenfest’s adiabatic principle, which demonstrated a close connection between classical dynamics and quantization conditions, Bohr formalized a technique that he had used implicitly in his 1913 model of hydrogen, and now elevated it to the status of a fundamental principle of quantum theory.  He called it the Correspondence Principle, and published the details in 1920.  The Correspondence Principle states that as the quantum number of an electron orbit increases to large values, the quantum behavior converges to classical behavior.  Specifically, if an electron in a state of high quantum number emits a photon while jumping to a neighboring orbit, then the wavelength of the emitted photon approaches the classical radiation wavelength of the electron subject to Maxwell’s equations. 

Bohr’s Correspondence Principle cemented the bridge between classical physics and quantum physics.  One of the biggest former questions about the physics of electron orbits in atoms was why they did not radiate continuously because of the angular acceleration they experienced in their orbits.  Bohr had now reconnected to Maxwell’s equations and classical physics in the limit.  Like the theory of adiabatic invariants, the Correspondence Principle became a new tool for distinguishing among different quantum theories.  It could be used as a filter to distinguish “correct” quantum models, that transitioned smoothly from quantum to classical behavior, from those that did not.  Bohr’s Correspondence Principle was to be a powerful tool in the hands of Werner Heisenberg as he reinvented quantum theory only a few years later.

Quantization conditions.

 By the end of 1920, all the elements of the quantum theory of electron orbits were apparently falling into place.  Bohr’s originally ad hoc quantization condition was now on firm footing.  The quantization conditions were related to action integrals that were, in turn, adiabatic invariants of the classical dynamics.  This meant that slight variations in the parameters of the dynamics systems would not induce quantum transitions among the various quantum states.  This conclusion would have felt right to the early quantum practitioners.  Bohr’s quantum model of electron orbits was fundamentally a means of explaining quantum transitions between stationary states.  Now it appeared that the condition for the stationary states of the electron orbits was an insensitivity, or invariance, to variations in the dynamical properties.  This was analogous to the principle of stationary action where the action along a dynamical trajectory is invariant to slight variations in the trajectory.  Therefore, the theory of quantum orbits now rested on firm foundations that seemed as solid as the foundations of classical mechanics.

From the perspective of modern quantum theory, the concept of elliptical Keplerian orbits for the electron is grossly inaccurate.  Most physicists shudder when they see the symbol for atomic energy—the classic but mistaken icon of electron orbits around a nucleus.  Nonetheless, Bohr and Ehrenfest and Sommerfeld had hit on a deep thread that runs through all of physics—the concept of action—the same concept that Leibniz introduced, that Maupertuis minimized and that Euler canonized.  This concept of action is at work in the macroscopic domain of classical dynamics as well as the microscopic world of quantum phenomena.  Planck was acutely aware of this connection with action, which is why he so readily recognized his elementary constant as the quantum of action. 

However, the old quantum theory was running out of steam.  For instance, the action integrals and adiabatic invariants only worked for single electron orbits, leaving the vast bulk of many-electron atomic matter beyond the reach of quantum theory and prediction.  The literal electron orbits were a crutch or bias that prevented physicists from moving past them and seeing new possibilities for quantum theory.  Orbits were an anachronism, exerting a damping force on progress.  This limitation became painfully clear when Bohr and his assistants at Copenhagen–Kramers and Slater–attempted to use their electron orbits to explain the refractive index of gases.  The theory was cumbersome and exhausted.  It was time for a new quantum revolution by a new generation of quantum wizards–Heisenberg, Born, Schrödinger, Pauli, Jordan and Dirac.


[1] N. Bohr, “On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus,” Philosophical Magazine, vol. 26, pp. 476–502, 1913.

[2] A. Sommerfeld, “The quantum theory of spectral lines,” Annalen Der Physik, vol. 51, pp. 1-94, Sep 1916.

[3] P. Ehrenfest, “Een mechanische theorema van Boltzmann en zijne betrekking tot de quanta theorie (A mechanical theorem of Boltzmann and its relation to the theory of energy quanta),” Verslag van de Gewoge Vergaderingen der Wis-en Natuurkungige Afdeeling, vol. 22, pp. 586-593, 1913.

[4] P. Ehrenfest, “Adiabatic invariables and quantum theory,” Annalen Der Physik, vol. 51, pp. 327-352, Oct 1916.