How Number Theory Protects You from the Chaos of the Cosmos

We are exceedingly fortunate that the Earth lies in the Goldilocks zone.  This zone is the range of orbital radii of a planet around its sun for which water can exist in a liquid state.  Water is the universal solvent, and it may be a prerequisite for the evolution of life.  If we were too close to the sun, water would evaporate as steam.  And if we are too far, then it would be locked in perpetual ice.  As it is, the Earth has had wild swings in its surface temperature.  There was once a time, more than 650 million years ago, when the entire Earth’s surface froze over.  Fortunately, the liquid oceans remained liquid, and life that already existed on Earth was able to persist long enough to get to the Cambrian explosion.  Conversely, Venus may once have had liquid oceans and maybe even nascent life, but too much carbon dioxide turned the planet into an oven and boiled away its water (a fate that may await our own Earth if we aren’t careful).  What has saved us so far is the stability of our orbit, our steady distance from the Sun that keeps our water liquid and life flourishing.  Yet it did not have to be this way. 

The regions of regular motion associated with irrational numbers act as if they were a barrier, restricting the range of chaotic orbits and protecting other nearby orbits from the chaos.

Our solar system is a many-body problem.  It consists of three large gravitating bodies (Sun, Jupiter, Saturn) and several minor ones (such as Earth).   Jupiter does influence our orbit, and if it were only a few times more massive than it actually is, then our orbit would become chaotic, varying in distance from the sun in unpredictable ways.  And if Jupiter were only about 20 times bigger than is actually is, there is a possibility that it would perturb the Earth’s orbit so strongly that it could eject the Earth from the solar system entirely, sending us flying through interstellar space, where we would slowly cool until we became a permanent ice ball.  What can protect us from this terrifying fate?  What keeps our orbit stable despite the fact that we inhabit a many-body solar system?  The answer is number theory!

The Most Irrational Number

What is the most irrational number you can think of? 

Is it: pi = 3.1415926535897932384626433 ? 

Or Euler’s constant: e = 2.7182818284590452353602874 ?

How about: sqrt(3) = 1.73205080756887729352744634 ?

These are all perfectly good irrational numbers.  But how do you choose the “most irrational” number?  The answer is fairly simple.  The most irrational number is the one that is least well approximated by a ratio of integers.  For instance, it is possible to get close to pi through the ratio 22/7 = 3.1428 which differs from pi by only 4 parts in ten thousand.  Or Euler’s constant 87/32 = 2.7188 differs from e by only 2 parts in ten thousand.  Yet 87 and 32 are much bigger than 22 and 7, so it may be said that e is more irrational than pi, because it takes ratios of larger integers to get a good approximation.  So is there a “most irrational” number?  The answer is yes.  The Golden Ratio.

The Golden ratio can be defined in many ways, but its most common expression is given by

It is the hardest number to approximate with a ratio of small integers.  For instance, to get a number that is as close as one part in ten thousand to the golden mean takes the ratio 89/55.  This result may seem obscure, but there is a systematic way to find the ratios of integers that approximate an irrational number. This is known as a convergent from continued fractions.

Continued fractions were invented by John Wallis in 1695, introduced in his book Opera Mathematica.  The continued fraction for pi is

An alternate form of displaying this continued fraction is with the expression

The irrational character of pi is captured by the seemingly random integers in this string. However, there can be regular structure in irrational numbers. For instance, a different continued fraction for pi is

that has a surprisingly simple repeating pattern.

The continued fraction for the golden mean has an especially simple repeating form

or

This continued fraction has the slowest convergence for its continued fraction of any other number. Hence, the Golden Ratio can be considered, using this criterion, to be the most irrational number.

If the Golden Ratio is the most irrational number, how does that save us from the chaos of the cosmos? The answer to this question is KAM!

Kolmogorov, Arnold and Moser: (KAM) Theory

KAM is an acronym made from the first initials of three towering mathematicians of the 20th century: Andrey Kolmogorov (1903 – 1987), his student Vladimir Arnold (1937 – 2010), and Jürgen Moser (1928 – 1999).

In 1954, Kolmogorov, considered to be the greatest living mathematician at that time, was invited to give the plenary lecture at a mathematics conference. To the surprise of the conference organizers, he chose to talk on what seemed like a very mundane topic: the question of the stability of the solar system. This had been the topic which Poincaré had attempted to solve in 1890 when he first stumbled on chaotic dynamics. The question had remained open, but the general consensus was that the many-body nature of the solar system made it intrinsically unstable, even for only three bodies.

Against all expectations, Kolmogorov proposed that despite the general chaotic behavior of the three–body problem, there could be “islands of stability” which were protected from chaos, allowing some orbits to remain regular even while other nearby orbits were highly chaotic. He even outlined an approach to a proof of his conjecture, though he had not carried it through to completion.

The proof of Kolmogorov’s conjecture was supplied over the next 10 years through the work of the German mathematician Jürgen Moser and by Kolmogorov’s former student Vladimir Arnold. The proof hinged on the successive ratios of integers that approximate irrational numbers. With this work KAM showed that indeed some orbits are actually protected from neighboring chaos by relying on the irrationality of the ratio of orbital periods.

Resonant Ratios

Let’s go back to the simple model of our solar system that consists of only three bodies: the Sun, Jupiter and Earth. The period of Jupiter’s orbit is 11.86 years, but instead, if it were exactly 12 years, then its period would be in a 12:1 ratio with the Earth’s period. This ratio of integers is called a “resonance”, although in this case it is fairly mismatched. But if this ratio were a ratio of small integers like 5:3, then it means that Jupiter would travel around the sun 5 times in 15 years while the Earth went around 3 times. And every 15 years, the two planets would align. This kind of resonance with ratios of small integers creates a strong gravitational perturbation that alters the orbit of the smaller planet. If the perturbation is strong enough, it could disrupt the Earth’s orbit, creating a chaotic path that might ultimately eject the Earth completely from the solar system.

What KAM discovered is that as the resonance ratio becomes a ratio of large integers, like 87:32, then the planets have a hard time aligning, and the perturbation remains small. A surprising part of this theory is that a nearby orbital ratio might be 5:2 = 1.5, which is only a little different than 87:32 = 1.7. Yet the 5:2 resonance can produce strong chaos, while the 87:32 resonance is almost immune. This way, it is possible to have both chaotic orbits and regular orbits coexisting in the same dynamical system. An irrational orbital ratio protects the regular orbits from chaos. The next question is, how irrational does the orbital ratio need to be to guarantee safety?

You probably already guessed the answer to this question–the answer must be the Golden Ratio. If this is indeed the most irrational number, then it cannot be approximated very well with ratios of small integers, and this is indeed the case. In a three-body system, the most stable orbital ratio would be a ratio of 1.618034. But the more general question of what is “irrational enough” for an orbit to be stable against a given perturbation is much harder to answer. This is the field of Diophantine Analysis, which addresses other questions as well, such as Fermat’s Last Theorem.

KAM Twist Map

The dynamics of three-body systems are hard to visualize directly, so there are tricks that help bring the problem into perspective. The first trick, invented by Henri Poincaré, is called the first return map (or the Poincaré section). This is a way of reducing the dimensionality of the problem by one dimension. But for three bodies, even if they are all in a plane, this still can be complicated. Another trick, called the restricted three-body problem, is to assume that there are two large masses and a third small mass. This way, the dynamics of the two-body system is unaffected by the small mass, so all we need to do is focus on the dynamics of the small body. This brings the dynamics down to two dimensions (the position and momentum of the third body), which is very convenient for visualization, but the dynamics still need solutions to differential equations. So the final trick is to replace the differential equations with simple difference equations that are solved iteratively.

A simple discrete iterative map that captures the essential behavior of the three-body problem begins with action-angle variables that are coupled through a perturbation. Variations on this model have several names: the Twist Map, the Chirikov Map and the Standard Map. The essential mapping is

where J is an action variable (like angular momentum) paired with the angle variable. Initial conditions for the action and the angle are selected, and then all later values are obtained by iteration. The perturbation parameter is given by ε. If ε = 0 then all orbits are perfectly regular and circular. But as the perturbation increases, the open orbits split up into chains of closed (periodic) orbits. As the perturbation increases further, chaotic behavior emerges. The situation for ε = 0.9 is shown in the figure below. There are many regular periodic orbits as well as open orbits. Yet there are simultaneously regions of chaotic behavior. This figure shows an intermediate case where regular orbits can coexist with chaotic ones. The key is the orbital period ratio. For orbital ratios that are sufficiently irrational, the orbits remain open and regular. Bur for orbital ratios that are ratios of small integers, the perturbation is strong enough to drive the dynamics into chaos.

Arnold Twist Map (also known as a Chirikov map) for ε = 0.9 showing the chaos that has emerged at the hyperbolic point, but there are still open orbits that are surprisingly circular (unperturbed) despite the presence of strongly chaotic orbits nearby.

Python Code

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Oct. 2, 2019
@author: nolte
"""
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
plt.close('all')

eps = 0.9

np.random.seed(2)
plt.figure(1)
for eloop in range(0,50):

    rlast = np.pi*(1.5*np.random.random()-0.5)
    thlast = 2*np.pi*np.random.random()

    orbit = np.int(200*(rlast+np.pi/2))
    rplot = np.zeros(shape=(orbit,))
    thetaplot = np.zeros(shape=(orbit,))
    x = np.zeros(shape=(orbit,))
    y = np.zeros(shape=(orbit,))    
    for loop in range(0,orbit):
        rnew = rlast + eps*np.sin(thlast)
        thnew = np.mod(thlast+rnew,2*np.pi)
        
        rplot[loop] = rnew
        thetaplot[loop] = np.mod(thnew-np.pi,2*np.pi) - np.pi            
          
        rlast = rnew
        thlast = thnew
        
        x[loop] = (rnew+np.pi+0.25)*np.cos(thnew)
        y[loop] = (rnew+np.pi+0.25)*np.sin(thnew)
        
    plt.plot(x,y,'o',ms=1)

plt.savefig('StandMapTwist')

The twist map for three values of ε are shown in the figure below. For ε = 0.2, most orbits are open, with one elliptic point and its associated hyperbolic point. At ε = 0.9 the periodic elliptic point is still stable, but the hyperbolic point has generated a region of chaotic orbits. There is still a remnant open orbit that is associated with an orbital period ratio at the Golden Ratio. However, by ε = 0.97, even this most stable orbit has broken up into a chain of closed orbits as the chaotic regions expand.

Twist map for three levels of perturbation.

Safety in Numbers

In our solar system, governed by gravitational attractions, the square of the orbital period increases as the cube of the average radius (Kepler’s third law). Consider the restricted three-body problem of the Sun and Jupiter with the Earth as the third body. If we analyze the stability of the Earth’s orbit as a function of distance from the Sun, the orbital ratio relative to Jupiter would change smoothly. Near our current position, it would be in a 12:1 resonance, but as we moved farther from the Sun, this ratio would decrease. When the orbital period ratio is sufficiently irrational, then the orbit would be immune to Jupiter’s pull. But as the orbital ratio approaches ratios of integers, the effect gets larger. Close enough to Jupiter there would be a succession of radii that had regular motion separated by regions of chaotic motion. The regions of regular motion associated with irrational numbers act as if they were a barrier, restricting the range of chaotic orbits and protecting more distant orbits from the chaos. In this way numbers, rational versus irrational, protect us from the chaos of our own solar system.

A dramatic demonstration of the orbital resonance effect can be seen with the asteroid belt. The many small bodies act as probes of the orbital resonances. The repetitive tug of Jupiter opens gaps in the distribution of asteroid radii, with major gaps, called Kirkwood Gaps, opening at orbital ratios of 3:1, 5:2, 7:3 and 2:1. These gaps are the radii where chaotic behavior occurs, while the regions in between are stable. Most asteroids spend most of their time in the stable regions, because chaotic motion tends to sweep them out of the regions of resonance. This mechanism for the Kirkwood gaps is the same physics that produces gaps in the rings of Saturn at resonances with the many moons of Saturn.

The gaps in the asteroid distributions caused by orbital resonances with Jupiter. Ref. Wikipedia

Further Reading

For a detailed history of the development of KAM theory, see Chapter 9 Butterflies to Hurricanes in Galileo Unbound (Oxford University Press, 2018).

For a more detailed mathematical description of the KAM theory, see Chapter 5, Hamiltonian Chaos, in Introduction to Modern Dynamics, 2nd edition (Oxford University Press, 2019).

See also:

Dumas, H. S., The KAM Story: A friendly introduction to the content, history and significance of Classical Kolmogorov-Arnold-Moser Theory. World Scientific: 2014.

Arnold, V. I., From superpositions to KAM theory. Vladimir Igorevich Arnold. Selected Papers 1997, PHASIS, 60, 727–740.

Vladimir Arnold’s Cat Map

The 1960’s are known as a time of cultural revolution, but perhaps less known was the revolution that occurred in the science of dynamics.  Three towering figures of that revolution were Stephen Smale (1930 – ) at Berkeley, Andrey Kolmogorov (1903 – 1987) in Moscow and his student Vladimir Arnold (1937 – 2010).  Arnold was only 20 years old in 1957 when he solved Hilbert’s thirteenth problem (that any continuous function of several variables can be constructed with a finite number of two-variable functions).  Only a few years later his work on the problem of small denominators in dynamical systems provided the finishing touches on the long elusive explanation of the stability of the solar system (the problem for which Poincaré won the King Oscar Prize in mathematics in 1889 when he discovered chaotic dynamics ).  This theory is known as KAM-theory, using the first initials of the names of Kolmogorov, Arnold and Moser [1].  Building on his breakthrough in celestial mechanics, Arnold’s work through the 1960’s remade the theory of Hamiltonian systems, creating a shift in perspective that has permanently altered how physicists look at dynamical systems.

Hamiltonian Physics on a Torus

Traditionally, Hamiltonian physics is associated with systems of inertial objects that conserve the sum of kinetic and potential energy, in other words, conservative non-dissipative systems.  But a modern view (after Arnold) of Hamiltonian systems sees them as hyperdimensional mathematical mappings that conserve volume.  The space that these mappings inhabit is phase space, and the conservation of phase-space volume is known as Liouville’s Theorem [2].  The geometry of phase space is called symplectic geometry, and the universal position that symplectic geometry now holds in the physics of Hamiltonian mechanics is largely due to Arnold’s textbook Mathematical Methods of Classical Mechanics (1974, English translation 1978) [3]. Arnold’s famous quote from that text is “Hamiltonian mechanics is geometry in phase space”. 

One of the striking aspects of this textbook is the reduction of phase-space geometry to the geometry of a hyperdimensional torus for a large number of Hamiltonian systems.  If there are as many conserved quantities as there are degrees of freedom in a Hamiltonian system, then the system is called “integrable” (because you can integrated the equations of motion to find a constant of the motion). Then it is possible to map the physics onto a hyperdimensional torus through the transformation of dynamical coordinates into what are known as “action-angle” coordinates [4].  Each independent angle has an associated action that is conserved during the motion of the system.  The periodicity of the dynamical angle coordinate makes it possible to identify it with the angular coordinate of a multi-dimensional torus.  Therefore, every integrable Hamiltonian system can be mapped to motion on a multi-dimensional torus (one dimension for each degree of freedom of the system). 

Actually, integrable Hamiltonian systems are among the most boring dynamical systems you can imagine. They literally just go in circles (around the torus). But as soon as you add a small perturbation that cannot be integrated they produce some of the most complex and beautiful patterns of all dynamical systems. It was Arnold’s focus on motions on a torus, and perturbations that shift the dynamics off the torus, that led him to propose a simple mapping that captured the essence of Hamiltonian chaos.

The Arnold Cat Map

Motion on a two-dimensional torus is defined by two angles, and trajectories on a two-dimensional torus are simple helixes. If the periodicities of the motion in the two angles have an integer ratio, the helix repeats itself. However, if the ratio of periods (also known as the winding number) is irrational, then the helix never repeats and passes arbitrarily closely to any point on the surface of the torus. This last case leads to an “ergodic” system, which is a term introduced by Boltzmann to describe a physical system whose trajectory fills phase space. The behavior of a helix for rational or irrational winding number is not terribly interesting. It’s just an orbit going in circles like an integrable Hamiltonian system. The helix can never even cross itself.

However, if you could add a new dimension to the torus (or add a new degree of freedom to the dynamical system), then the helix could pass over or under itself by moving into the new dimension. By weaving around itself, a trajectory can become chaotic, and the set of many trajectories can become as mixed up as a bowl of spaghetti. This can be a little hard to visualize, especially in higher dimensions, but Arnold thought of a very simple mathematical mapping that captures the essential motion on a torus, preserving volume as required for a Hamiltonian system, but with the ability for regions to become all mixed up, just like trajectories in a nonintegrable Hamiltonian system.

A unit square is isomorphic to a two-dimensional torus. This means that there is a one-to-one mapping of each point on the unit square to each point on the surface of a torus. Imagine taking a sheet of paper and forming a tube out of it. One of the dimensions of the sheet of paper is now an angle coordinate that is cyclic, going around the circumference of the tube. Now if the sheet of paper is flexible (like it is made of thin rubber) you can bend the tube around and connect the top of the tube with the bottom, like a bicycle inner tube. The other dimension of the sheet of paper is now also an angle coordinate that is cyclic. In this way a flat sheet is converted (with some bending) into a torus.

Arnold’s key idea was to create a transformation that takes the torus into itself, preserving volume, yet including the ability for regions to pass around each other. Arnold accomplished this with the simple map

where the modulus 1 takes the unit square into itself. This transformation can also be expressed as a matrix

followed by taking modulus 1. The transformation matrix is called a Floquet matrix, and the determinant of the matrix is equal to unity, which ensures that volume is conserved.

Arnold decided to illustrate this mapping by using a crude image of the face of a cat (See Fig. 1). Successive applications of the transformation stretch and shear the cat, which is then folded back into the unit square. The stretching and folding preserve the volume, but the image becomes all mixed up, just like mixing in a chaotic Hamiltonian system, or like an immiscible dye in water that is stirred.

Fig. 1 Arnold’s illustration of his cat map from pg. 6 of V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, 1968) [5]
Fig. 2 Arnold Cat Map operation is an iterated succession of stretching with shear of a unit square, and translation back to the unit square. The mapping preserves and mixes areas, and is invertible.

Recurrence

When the transformation matrix is applied to continuous values, it produces a continuous range of transformed values that become thinner and thinner until the unit square is uniformly mixed. However, if the unit square is discrete, made up of pixels, then something very different happens (see Fig. 3). The image of the cat in this case is composed of a 50×50 array of pixels. For early iterations, the image becomes stretched and mixed, but at iteration 50 there are 4 low-resolution upside-down versions of the cat, and at iteration 75 the cat fully reforms, but is upside-down. Continuing on, the cat eventually reappears fully reformed and upright at iteration 150. Therefore, the discrete case displays a recurrence and the mapping is periodic. Calculating the period of the cat map on lattices can lead to interesting patterns, especially if the lattice is composed of prime numbers [6].

Fig. 3 A discrete cat map has a recurrence period. This example with a 50×50 lattice has a period of 150.

The Cat Map and the Golden Mean

The golden mean, or the golden ratio, 1.618033988749895 is never far away when working with Hamiltonian systems. Because the golden mean is the “most irrational” of all irrational numbers, it plays an essential role in KAM theory on the stability of the solar system. In the case of Arnold’s cat map, it pops up its head in several ways. For instance, the transformation matrix has eigenvalues

with the remarkable property that

which guarantees conservation of area.


Selected V. I. Arnold Publications

Arnold, V. I. “FUNCTIONS OF 3 VARIABLES.” Doklady Akademii Nauk Sssr 114(4): 679-681. (1957)

Arnold, V. I. “GENERATION OF QUASI-PERIODIC MOTION FROM A FAMILY OF PERIODIC MOTIONS.” Doklady Akademii Nauk Sssr 138(1): 13-&. (1961)

Arnold, V. I. “STABILITY OF EQUILIBRIUM POSITION OF A HAMILTONIAN SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS IN GENERAL ELLIPTIC CASE.” Doklady Akademii Nauk Sssr 137(2): 255-&. (1961)

Arnold, V. I. “BEHAVIOUR OF AN ADIABATIC INVARIANT WHEN HAMILTONS FUNCTION IS UNDERGOING A SLOW PERIODIC VARIATION.” Doklady Akademii Nauk Sssr 142(4): 758-&. (1962)

Arnold, V. I. “CLASSICAL THEORY OF PERTURBATIONS AND PROBLEM OF STABILITY OF PLANETARY SYSTEMS.” Doklady Akademii Nauk Sssr 145(3): 487-&. (1962)

Arnold, V. I. “BEHAVIOUR OF AN ADIABATIC INVARIANT WHEN HAMILTONS FUNCTION IS UNDERGOING A SLOW PERIODIC VARIATION.” Doklady Akademii Nauk Sssr 142(4): 758-&. (1962)

Arnold, V. I. and Y. G. Sinai. “SMALL PERTURBATIONS OF AUTHOMORPHISMS OF A TORE.” Doklady Akademii Nauk Sssr 144(4): 695-&. (1962)

Arnold, V. I. “Small denominators and problems of the stability of motion in classical and celestial mechanics (in Russian).” Usp. Mat. Nauk. 18: 91-192. (1963)

Arnold, V. I. and A. L. Krylov. “UNIFORM DISTRIBUTION OF POINTS ON A SPHERE AND SOME ERGODIC PROPERTIES OF SOLUTIONS TO LINEAR ORDINARY DIFFERENTIAL EQUATIONS IN COMPLEX REGION.” Doklady Akademii Nauk Sssr 148(1): 9-&. (1963)

Arnold, V. I. “INSTABILITY OF DYNAMICAL SYSTEMS WITH MANY DEGREES OF FREEDOM.” Doklady Akademii Nauk Sssr 156(1): 9-&. (1964)

Arnold, V. “SUR UNE PROPRIETE TOPOLOGIQUE DES APPLICATIONS GLOBALEMENT CANONIQUES DE LA MECANIQUE CLASSIQUE.” Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences 261(19): 3719-&. (1965)

Arnold, V. I. “APPLICABILITY CONDITIONS AND ERROR ESTIMATION BY AVERAGING FOR SYSTEMS WHICH GO THROUGH RESONANCES IN COURSE OF EVOLUTION.” Doklady Akademii Nauk Sssr 161(1): 9-&. (1965)


Bibliography

[1] Dumas, H. S. The KAM Story: A friendly introduction to the content, history and significance of Classical Kolmogorov-Arnold-Moser Theory, World Scientific. (2014)

[2] See Chapter 6, “The Tangled Tale of Phase Space” in Galileo Unbound (D. D. Nolte, Oxford University Press, 2018)

[3] V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauk 1974, English translation Springer 1978)

[4] See Chapter 3, “Hamiltonian Dynamics and Phase Space” in Introduction to Modern Dynamics, 2nd ed. (D. D. Nolte, Oxford University Press, 2019)

[5] V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, 1968)

[6] Gaspari, G. “THE ARNOLD CAT MAP ON PRIME LATTICES.” Physica D-Nonlinear Phenomena 73(4): 352-372. (1994)

Top 10 Topics of Modern Dynamics

“Modern physics” in the undergraduate physics curriculum has been monopolized, on the one hand, by quantum mechanics, nuclear physics, particle physics and astrophysics. “Classical mechanics”, on the other hand, has been monopolized by Lagrangians and Hamiltonians.  While these are all admittedly interesting, the topics of modern dynamics that monopolize the time and actions of most physics-degree holders, as they work in high-tech start-ups, established technology companies, or on Wall Street, are not to be found.  These are the topics of nonlinear dynamics, chaos theory, complex networks, finance, evolutionary dynamics and neural networks, among others.

Cover

There is a growing awareness that the undergraduate physics curriculum needs to be reinvigorated to make a physics degree relevant to the modern workplace.  To that end, I am listing my top 10 topics of modern dynamics that can form the foundation of a revamped upper-division (junior level) mechanics course.  Virtually all of these topics were once reserved for graduate-student-level courses, but all can be introduced to undergraduates in simple and intuitive ways without the need for advanced math.

1) Phase Space

The key change in perspective for modern dynamics that differentiates it from classical dynamics is the emphasis on the set of all possible trajectories that fill a “space” rather than emphasizing single trajectories defined by given initial conditions.  Rather than study the motion of one rock thrown from a cliff top, modern dynamics studies an infinity of rocks thrown from every possible point and with every possible velocity.  The space that contains this infinity of trajectories is known as phase space (or more generally state space).  The equation of motion in state space becomes the dynamical flow, replacing Newton’s second law as the central mathematical structure of physics.  Modern dynamics studies the properties of phase space rather than the properties of single trajectories, and makes rigorous and unique conclusions about classes of possible motions.  This emphasis on classes of behavior is more general and more universal and more powerful, while also providing a fundamental “visual language” with which to describe the complex physics of complex systems.

2) Metric Space

The Cartesian coordinate plane that we were all taught in high school tends to dominate our thinking, biasing us towards linear flat geometries.  Yet most dynamics do not take place in such simple Cartesian spaces.  A case in point, virtually every real-world dynamics problem has constraints that confine the motion to a surface.  Furthermore, the number of degrees of freedom of a dynamical system usually exceed our common 3-space, expanding to hundreds or even to thousands of dimensions.  The surfaces of constraint are hypersurfaces of high dimensions (known as manifolds) and are almost certainly not flat hyperplanes. This daunting prospect of high-dimensional warped spaces can be surprisingly simplified through the concept of Bernhard Riemann’s “metric space”.  Understanding the geometry of a metric space can be as simple as applying Pythagoras’ Theorem to sets of coordinates.  For instance, the metric tensor can be taught and used without requiring students to know anything of tensor calculus.  At the same time, it provides a useful tool for understanding dynamical patterns in phase space as well as orbits around black holes.

3) Invariants

Introductory physics classes emphasize the conservation of energy, linear momentum and angular momentum as if they are special cases.  Yet there is a grand structure that yields a universal set of conservation laws: integrable Hamiltonian systems.  An integrable system is one for which there are as many invariants of motion as there are degrees of freedom.  Amazingly, these conservation laws can all be captured by a single procedure known as (canonical) transformation to action-angle coordinates.  When expressed in action-angle form, these Hamiltonians take on extremely simple expressions.  They are also the starting point for the study of perturbations when small nonintegrable terms are added to the Hamiltonian.  As the perturbations grow, this provides one doorway to the emergence of chaos.

4) Chaos theory

“Chaos theory” is the more popular title for what is generally called “nonlinear dynamics”.  Nonlinear dynamics takes place in state space when the dynamical flow equations have terms that algebraically are products of variables.  One important distinction between chaos theory and nonlinear dynamics is the occurrence of unpredictability that can emerge in the dynamics when the number of variables is equal to three or higher.  The equations, and the resulting dynamics, are still deterministic, but the trajectories are incredibly sensitive to initial conditions (SIC).  In addition, the dynamical trajectories can relax to a submanifold of the original state space known as a strange attractor that typically is a fractal structure.

5) Synchronization

One of the central paradigms of nonlinear dynamics is the autonomous oscillator.  Unlike the harmonic oscillator that eventually decays due to friction, autonomous oscillators are steady-state oscillators that convert steady energy input into oscillatory behavior.  A prime example is the pendulum clock that converts the steady weight of a hanging mass into a sustained oscillation.  When two autonomous oscillators (that normally oscillator at slightly different frequencies) are coupled weakly together, they can synchronize to the same frequency.   This effect was discovered by Christiaan Huygens when he observed two pendulum clocks hanging next to each other on a wall synchronize the swings of their pendula.  Synchronization is a central paradigm in modern dynamics for several reasons.  First, it demonstrates the emergence of order when a collective behavior emerges from a collection of individual systems (this phenomenon of emergence is one of the fundamental principles of complex system science).  Second, synchronized systems include such critical systems as the beating heart and the thinking brain.  Third, synchronization becomes a useful tool to explore coupled systems that have a large number of linked subsystems, as in networks of nodes.

6) Network Dynamics

Networks have become one of the driving forces of our modern interconnected society.  The structure of networks, the dynamics of nodes in networks, and the dynamic growth of networks are all coming into focus as we live our lives in multiple interconnected webs.  Dynamics on networks include problems like diffusion and the spread of infection and connect with topics of percolation theory and critical phenomenon.  Nonlinear dynamics on networks provide key opportunities and examples to study complex interacting systems.

7) Neural Networks

Perhaps the most enigmatic network is the network of neurons in the brain.  The emergence of intelligence and of sentience is one of the greatest scientific questions.  At a much simpler level, the nonlinear dynamics of small numbers of neurons display the properties of autonomous oscillators and synchronization, while larger sets of neurons become interconnected into dynamic networks.  The dynamics of neurons and of neural networks is a  key topic in modern dynamics.  Not only can the physics of the networks be studied, but neural networks become tools for studying other complex systems.

8) Evolutionary Dynamics

The emergence of life and the evolution of species stands as another of the greatest scientific questions of our day.  Although this topic traditionally is studied by the biological sciences (and mathematical biology), physics has a surprising lot to say on the topic.  The dynamics of evolution can be captured in the same types of nonlinear flows that live in state space.  For instance, population dynamics can be described as a large ensemble of interacting individuals that are born, flourish and die dependent on their environment and on their complicated interactions with other members in their ecosystem.  These types of problems have state spaces of extremely high dimension far beyond what we can visualize.  Yet the emergence of structure and of patterns from the complex dynamics helps to reduce the complexity, as do conceptual metaphors like evolutionary fitness landscapes.

9) Economic Dynamics

A non-negligible fraction of both undergraduate and graduate physics degree holders end up on Wall Street or in related industries.  This is partly because physicists are numerically fluent while also possessing sound intuition.  Therefore, economic dynamics is a potentially valuable addition to the modern dynamics curriculum and easily expressed using the concepts of dynamical flows and state space.  Both microeconomics (business competition, business cycles) and macroeconomics (investment and savings, liquidity and money, inflation, unemployment) can be described and analyzed using mathematical flows that are the central toolkit of modern dynamics.

10) Relativity

Special relativity is a common topic in the current upper-division physics curriculum, while general relativity is viewed as too difficult to expose undergraduates to.  This is mostly an artificial division, because Einstein’s “happiest thought” occurred when he realized that an observer in free fall is in a force-free (inertial) frame.  The equivalence principle, that states that a frame in uniform acceleration is indistinguishable from a stationary frame in a uniform gravitational field, opens a wide door that connects special relativity to general relativity.  In an undergraduate course on modern dynamics, the metric tensor (described above) is introduced in simple terms, providing the foundation to develop Minkowski spacetime, and the next natural extension is to warped spacetime—all at the simple level of linear algebra combined with partial differentiation.  General relativity ties in many of the principles of the modern dynamics curriculum (dynamical flows, state space, metric space, invariants, nonlinear dynamics), and the students can simulate orbits around black holes with ease.  I have been teaching General Relativity to undergraduates for over ten years now, and it is a highlight of the course.

Introduction to Modern Dynamics

For further reading and more details, these top 10 topics of modern dynamics are defined and explored in the undergraduate physics textbook “Introduction to Modern Dynamics: Chaos, Networks, Space and Time” published by Oxford University Press (2015).  This textbook is designed for use in a two-semester junior-level mechanics course.  It introduces the topics of modern dynamics, while still presenting traditional materials that the students need for their physics GREs.