Posts by David D. Nolte

E. M. Purcell Distinguished Professor of Physics and Astronomy at Purdue University

Huygens’ Tautochrone

In February of 1662, Pierre de Fermat wrote a paper Synthesis ad refractiones that explained Descartes-Snell’s Law of light refraction by finding the least time it took for light to travel between two points. This famous approach is now known as Fermat’s principle, and it motivated other searches for minimum principles. A few years earlier, in 1656, Christiaan Huygens had invented the pendulum clock [1], and he began a ten-year study of the physics of the pendulum. He was well aware that the pendulum clock does not keep exact time—as the pendulum swings wider, the period of oscillation slows down. He began to search for a path of the pendular mass that would keep the period the same (and make pendulum clocks more accurate), and he discovered a trajectory along which a mass would arrive at the same position in the same time no matter where it was released on the curve. That such a curve could exist was truly remarkable, and it promised to make highly accurate time pieces.

It made minimization problems a familiar part of physics—they became part of the mindset, leading ultimately to the principle of least action.

This curve is known as a tautochrone (literally: same or equal time) and Huygens provided a geometric proof in his Horologium Oscillatorium sive de motu pendulorum (1673) that the curve was a cycloid. A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls without slipping along a straight line. Huygens invented such a pendulum in which the mass executed a cycloid curve. It was a mass on a flexible yet inelastic string that partially wrapped itself around a solid bumper on each half swing. In principle, whether the pendulum swung gently, or through large displacements, the time would be the same. Unfortunately, friction along the contact of the string with the bumper prevented the pendulum from achieving this goal, and the tautochronic pendulum did not catch on.


Fig. 1 Huygens’ isochronous pendulum.  The time it takes the pendulum bob to follow the cycloid arc is independent of the pendulum’s amplitude, unlike for the circular arc, as the pendulum slows down for larger excursions.

The solution of the tautochrone curve of equal time led naturally to a search for the curve of least time, known as the brachistochrone curve for a particle subject to gravity, like a bead sliding on a frictionless wire between two points. Johann Bernoulli published a challenge to find the brachistochrone in 1696 in the scientific journal Acta Eruditorum that had been founded in 1682 by Leibniz in Germany in collaboration with Otto Mencke. Leibniz envisioned the journal to be a place where new ideas in the natural sciences and mathematics could be published and disseminated rapidly, and it included letters and commentaries, acting as a communication hub to help establish a community of scholars across Europe. In reality, it was the continental response to the Proceedings of the Royal Society in England.  Naturally, the Acta and the Proceedings would later take partisan sides in the priority dispute between Leibniz and Newton for the development of the calculus.

When Bernoulli published his brachistochrone challenge in the June issue of 1696, it was read immediately by the leading mathematicians of the day, many of whom took up the challenge and replied. The problem was solved and published in the May 1697 issue of the Acta by no less than five correspondents, including Johann Bernoulli, Jakob Bernoulli (Johann’s brother), Isaac Newton, Gottfried Leibniz and Ehrenfried Walther von Tschirnhaus. Each of them varied in their approaches, but all found the same solution. Johann and Jakob each considered the problem as the path of a light beam in a medium whose speed varied with depth. Just as in the tautochrone, the solved curve was a cycloid. The path of fastest time always started with a vertical path that allowed the fastest acceleration, and the point of greatest depth always was at the point of greatest horizontal speed.

The brachistrochrone problem led to the invention of the variational calculus, with first steps by Jakob Bernoulli and later more rigorous approaches by Euler.  However, its real importance is that it made minimization problems a familiar part of physics—they became part of the mindset, leading ultimately to the principle of least action.

[1] Galileo conceived of a pendulum clock in 1641, and his son Vincenzo started construction, but it was never finished.  Huygens submitted and received a patent in 1657 for a practical escape mechanism on pendulum clocks that is still used today.




Geometry as Motion

Nothing seems as static and as solid as geometry—there is even a subfield of geometry known as “solid geometry”. Geometric objects seem fixed in time and in space. Yet the very first algebraic description of geometry was born out of kinematic constructions of curves as René Descartes undertook the solution of an ancient Greek problem posed by Pappus of Alexandria (c. 290 – c. 350) that had remained unsolved for over a millennium. In the process, Descartes’ invented coordinate geometry.

Descartes used kinematic language in the process of drawing  curves, and he even talked about the speed of the moving point. In this sense, Descartes’ curves are trajectories.

The problem of Pappus relates to the construction of what were known as loci, or what today we call curves or functions. Loci are a smooth collection of points. For instance, the intersection of two fixed lines in a plane is a point. But if you allow one of the lines to move continuously in the plane, the intersection between the moving line and the fixed line sweeps out a continuous succession of points that describe a curve—in this case a new line. The problem posed by Pappus was to find the appropriate curve, or loci, when multiple lines are allowed to move continuously in the plane in such a way that their movements are related by given ratios. It can be shown easily in the case of two lines that the curves that are generated are other lines. As the number of lines increases to three or four lines, the loci become the conic sections: circle, ellipse, parabola and hyperbola. Pappus then asked what one would get if there were five such lines—what type of curves were these? This was the problem that attracted Descartes.

What Descartes did—the step that was so radical that it reinvented geometry—was to fix lines in position rather than merely in length. To us, in the 21st century, such an act appears so obvious as to remove any sense of awe. But by fixing a line in position, and by choosing a fixed origin on that line to which other points on the line were referenced by their distance from that origin, and other lines were referenced by their positions relative to the first line, then these distances could be viewed as unknown quantities whose solution could be sought through algebraic means. This was Descartes’ breakthrough that today is called “analytic geometry”— algebra could be used to find geometric properties.

Newton too viewed mathematical curves as living things that changed in time, which was one of the central ideas behind his fluxions—literally curves in flux.

Today, we would call the “locations” of the points their “coordinates”, and Descartes is almost universally credited with the discovery of the Cartesian coordinate system. Cartesian coordinates are the well-known grids of points, defined by the x-axis and the y-axis placed at right angles to each other, at whose intersection is the origin. Each point on the plane is defined by a pair of numbers, usually represented as (x, y). However, there are no grids or orthogonal axes in Descartes’ Géométrie, and there are no pairs of numbers defining locations of points. About the most Cartesian-like element that can be recognized in Descartes’ La Géométrie is the line of reference AB, as in Fig. 1.


Fig. 1 The first figure in Descartes’ Géométrie that defines 3 lines that are placed in position relative to the point marked A, which is the origin. The point C is one point on the loci that is to be found such that it satisfies given relationships to the 3 lines.


In his radical new approach to loci, Descartes used kinematic language in the process of drawing the curves, and he even talked about the speed of the moving point. In this sense, Descartes’ curves are trajectories, time-dependent things. Important editions of Descartes’ Discourse were published in two volumes in 1659 and 1661 which were read by Newton as a student at Cambridge. Newton also viewed mathematical curves as living things that changed in time, which was one of the central ideas behind his fluxions—literally curves in flux.


Descartes’ Odd Geometry

Rene Descartes was an unlikely candidate to revolutionize geometry. He began his career as a mercenary soldier, his mind wrapped around things like war and women, which are far from problems of existence and geometry. Descartes’ strange conversion from a life of action to a life of mind occurred on the night of November 10-11 in 1619 while he was bivouacked in an army encampment in Bavaria as a mercenary early in the Thirty Years’ War (1618—1648). On that night, Descartes dreamed that exact rational thought, even mathematical method, could be applied to problems of philosophy. This became his life’s work, and because he was a man of exceptional talent, he succeeded in exceptional ways.

Even Descartes’ footnotes were capable of launching new fields of thought.

Descartes left his mercenary employment and established himself in the free-thinking republic of the Netherlands which was ending the long process of casting off the yolk of Spanish rule towards the end of the Eighty Years War (1568—1648). In 1623, he settled in The Hague, a part of the republic that had been free of Spanish troops for many years, and after a brief absence (during which he witnessed the Siege of Rochelle by Cardinal Richelieu), he returned to the Netherlands in 1628, at the age of 32. He remained in the Netherlands, moving often, taking classes or teaching classes at the Universities of Leiden and Utrecht until 1649, when he was enticed away by Queen Christina of Sweden to colder climes and ultimately to his death.


Descartes’ original curve (AC), constructed on non-orthogonal (oblique) x and y coordinates (La Géométrie, 1637)

Descartes’ years in the Netherlands were the most productive epoch of his life as he created his philosophy and pursued his dream’s promise. He embarked on an ambitious project to codify his rational philosophy to gain a full understanding of natural philosophy. He called this work Treatise on the World, known in short as Le Monde, and it quite naturally adopted Copernicus’ heliocentric view of the solar system, which by that time had become widely accepted in learned circles even before Galileo’s publication in 1632 of his Dialogue Concerning the Two Chief World Systems. However, when Galileo was convicted in 1633 of suspicion of heresy (See Galileo Unbound, Oxford University Press, 2018), Descartes abruptly abandoned his plans to publish Le Monde, despite being in the Netherlands where he was well beyond the reach of the Church. It was, after all, the Dutch publisher Elzevir who published Galileo’s last work on the Two Sciences in 1638 when no Italian publishers would touch it. However, Descartes remained a devout Catholic throughout his life and had no desire to oppose its wishes. Despite this setback, Descartes continued to work on less controversial parts of his project, and in 1637 he published three essays preceded by a short introduction.

The introduction was called the Discourse on the Method (which contained his famous cogito ergo sum), and the three essays were La Dioptrique on optics, Les Météores on atmosphere and weather and finally La Géométrie on geometry in which he solved a problem posed by Pappus of Alexandria in the fourth century AD. Descartes sought to find a fundamental set of proven truths that would serve as the elements one could use in a deductive method to derive higher-level constructs. It was partially as an exercise in deductive reasoning that he sought to solve the classical mathematics problem posed by Pappus. La Géométrie was published as an essay following the much loftier Discourse, so even Descartes’ footnotes were capable of launching new fields of thought. The new field is called analytical geometry, also known as Cartesian or coordinate geometry, in which algebra is applied to geometric problems. Today, coordinates and functions are such natural elements of mathematics and physics, that it is odd to think that they emerged as demonstrations of abstract philosophy.

Bibliography:  R. Descartes, D. E. Smith, and M. L. Latham, The geometry of René Descartes. Chicago: Open Court Pub. Co., 1925.


The Oxford Scholars


Oxford University, and specifically Merton College, was a site of intense intellectual ferment in the middle of the Medieval Period around the time of Chaucer. A string of natural philosophers, today called the Oxford Scholars or the Oxford Calculators, began to explore early ideas of motion, taking the first bold steps beyond Aristotle. They were the first “physicists” (although that term would not be used until our own time) and laid the foundation upon which Galileo would build the first algebraic law of physics.

It is hard to imagine today what it was like doing mathematical physics in the fourteenth century. Mathematical symbolism did not exist in any form. Nothing stood for anything else, as we routinely use in algebra, and there were no equations, only proportions.

Thomas Bradwardine (1290 – 1349) was the first of the Scholars, arriving at Oxford around 1320. He came from a moderately wealthy family from Sussex on the southern coast of England not far from where the Anglo Saxon king Harold lost his kingdom and his life at the Battle of Hastings. The life of a scholar was not lucrative, so Bradwardine supported himself mainly through the royal patronage of Edward III, for whom he was chaplain and confessor during Edward’s campaigns in France, eventually becoming the Archbishop of Canterbury, although he died of the plague returning from Avignon before he could take up the position. When not campaigning or playing courtier, Bradwardine found time to develop a broad-ranging program of study that spanned from logic and theology to physics.


Merton College, Oxford (attribution: Andrew Shiva / Wikipedia)

Bradwardine began a reanalysis of an apparent paradox that stemmed from Aristotle’s theory of motion. As anyone with experience pushing a heavy box across a floor knows, the box does not move until sufficient force is applied. Today we say that the applied force must exceed the static force of friction. However, this everyday experience is at odds with Aristotle’s theory that placed motion in inverse proportion to the resistance. In this theory, only an infinite resistance could cause zero motion, yet the box does eventually move if enough force is applied. Bradwardine sought to resolve this paradox. Within the scholastic tradition, Aristotle was always assumed to have understood the truth, even if fourteenth-century scholars could not understand it themselves. Therefore, Bradwardine constructed a mathematical “explanation” that could preserve Aristotle’s theory of proportion while still accounting for the fact that the box does not move.

It is hard to imagine today what it was like doing mathematical physics in the fourteenth century. Mathematical symbolism did not exist in any form. Nothing stood for anything else, as we routinely use in algebra, and there were no equations, only proportions. The introduction of algebra into Europe through Arabic texts was a hundred years away. Not even Euclid or Archimedes had been rediscovered by Bradwardine’s day, so all he had to work with was Pythagorean theory of ratios and positive numbers—even negative numbers did not exist—and the only mathematical tools at his disposal were logic and language. Nonetheless, armed only with these sparse tools, Bradwardine constructed a verbal argument that the proportion of impressed force to resistance must itself be as a proportionality between speeds. As awkward as this sounds in words, it is a first intuitive step towards the concept of an exponential relationship—a power law. In Bradwardine’s rule for motion, the relationships among force, resistance and speed were like compounding interest on a loan. Bradwardine’s rule is not correct physics, because the exponential function is never zero, and because motion does not grow exponentially, but it did introduce the intuitive idea that a physical process could be nonlinear (using our modern terminology), changing from small effects to large effects disproportionate to the change in the cause. Therefore, the importance of Bradwardine was more his approach than his result. He applied mathematical reasoning to a problem of kinetics and set the stage for mathematical science.

A few years after Bradwardine had devised his rule of motion, a young mathematician named William Heytesbury (1313—1373) arrived as a fellow of Merton College. In the years that they overlapped at Oxford, one can only speculate what was transmitted from the senior to the junior fellow, but by 1335 Heytesbury had constructed a theory of continuous magnitudes and their continuous changes that included motion as a subset. The concept of the continuum had been a great challenge for Aristotelian theory, leading to many paradoxes or sophisms, like Zeno’s paradox that supposedly proved the impossibility of motion. Heytesbury shrewdly recognized that the problem was the ill-defined idea of instantaneous rate of change.

Heytesbury was just as handicapped as Bradwardine in his lack of mathematical notation, but worse, he was handicapped by the Aristotelian injunction against taking ratios of unlike qualities. According to Aristotle, proportions must only be taken of like qualities, such as one linear length to another, or one mass to another. To take a proportion of a mass to a length was nonsense. Today we call it “division” (more accurately a distribution), and mass divided by length is a linear mass density. Therefore, because speed is distance divided by time, no such ratio was possible in Heytesbury’s day because distance and time are unlike qualities. Heytesbury ingeniously got around this restriction by considering the linear distances covered by two moving objects in equal times. The resulting linear distances were similar qualities and could thereby be related by proportion. The ratio of the distances become the ratio of speeds, even though speed itself could not be defined directly. This was a first step, a new tool. Using this conceit, Heytesbury was able to go much farther, to grapple with the problem of nonuniform motion and hence the more general concept of instantaneous speed.

In the language of calculus (developed by Newton and Leibniz 300 years later), instantaneous speed is a ratio of an element of length to an element of time in the limit as the elements vanish uniformly. In the language of Heytesbury (Latin), instantaneous speed is simply the average speed between two neighboring speeds (still working with ratios of distances traversed in equal times). And those neighboring speeds are similarly the averages of their neighbors, until one reaches zero speed on one end and final speed on the other. Heytesbury called this kind of motion difform as opposed to uniform motion.

A special case of difform motion was uniformly difform motion—uniform acceleration. Acceleration was completely outside the grasp of Aristotelian philosophers, even Heytesbury, but he could imagine a speed that changed uniformly. This requires that the extra distance travelled during the succeeding time unit relative to the distance travelled during the current time unit has a fixed value. He then showed, without equations, using only his math-like language, that if a form changes uniformly in time (constant rate of change) then the average value of the form over a fixed time is equal to the average of the initial and final values. This work had a tremendous importance, not only for the history of mathematics, but also for the history of physics, because when the form in question is speed, then this represents the discovery of the mean speed theorem for the case of uniform acceleration. The mean speed theorem is often attributed to Galileo, who proved the theorem as part of his law of fall, and he deserves the attribution because there is an important difference in context. Heytesbury was not a scientist nor even a natural philosopher. He was a logician interested in sophisms that arose in discussions of Aristotle. The real purpose of Heytesbury’s analysis was to show that paradoxes like that of Zeno could be resolved within the Aristotelian system. He certainly was not thinking of falling bodies, whereas Galileo was.

Not long after Heytesbury demonstrated the mean speed theorem, he was joined at Merton College by yet another young fellow, Richard Swineshead (fl. 1340-1354). Bradwardine was already gone, but his reputation survived, as well as his memory, in the person of Heytesbury, and Swineshead became another member in the tradition of the Merton Scholars. He was perhaps the most adept at mathematics of the three, and he published several monumental treatises that mathematically expanded upon both Bradwardine and Heytesbury, systematizing their results and disseminating them in published accounts that spread across scholastic Europe—all still without formulas, symbols or equations. For these works, he became known as The Calculator. By consolidating and documenting the work of the Oxford Scholars, his influence on the subsequent history of thought was considerable, as he was widely read by later mathematicians, including Leibniz, who had a copy of Heytesbury in his personal library.

( To read more about the Oxford Scholars, and their connections with members of their contemporaries in Paris, see Chapter 3 of Galileo Unbound (Oxford University Press, 2018).)


A Wealth of Motions: Six Generations in the History of the Physics of Motion


Since Galileo launched his trajectory, there have been six broad generations that have traced the continuing development of concepts of motion. These are: 1) Universal Motion; 2) Phase Space; 3) Space-Time; 4) Geometric Dynamics; 5) Quantum Coherence; and 6) Complex Systems. These six generations were not all sequential, many evolving in parallel over the centuries, borrowing from each other, and there surely are other ways one could divide up the story of dynamics. But these six generations capture the grand concepts and the crucial paradigm shifts that are Galileo’s legacy, taking us from Galileo’s trajectory to the broad expanses across which physicists practice physics today.

Universal Motion emerged as a new concept when Isaac Newton proposed his theory of universal gravitation by which the force that causes apples to drop from trees is the same force that keeps the Moon in motion around the Earth, and the Earth in motion around the Sun. This was a bold step because even in Newton’s day, some still believed that celestial objects obeyed different laws. For instance, it was only through the work of Edmund Halley, a contemporary and friend of Newton’s, that comets were understood to travel in elliptical orbits obeying the same laws as the planets. Universal Motion included ideas of momentum from the start, while concepts of energy and potential, which fill out this first generation, took nearly a century to develop in the hands of many others, like Leibniz and Euler and the Bernoullis. This first generation was concluded by the masterwork of the Italian-French mathematician Joseph-Louis Lagrange, who also planted the seed of the second generation.

The second generation, culminating in the powerful and useful Phase Space, also took more than a century to mature. It began when Lagrange divorced dynamics from geometry, establishing generalized coordinates as surrogates to directions in space. Ironically, by discarding geometry, Lagrange laid the foundation for generalized spaces, because generalized coordinates could be anything, coming in any units and in any number, each coordinate having its companion velocity, doubling the dimension for every freedom. The Austrian physicist Ludwig Boltzmann expanded the number of dimensions to the scale of Avogadro’s number of particles, and he discovered the conservation of phase space volume, an invariance of phase space that stays the same even as 1023 atoms (Avogadro’s number) in ideal gases follow their random trajectories. The idea of phase space set the stage for statistical mechanics and for a new probabilistic viewpoint of mechanics that would extend into chaotic motions.

The French mathematician Henri Poincaré got a glimpse of chaotic motion in 1890 as he rushed to correct an embarrassing mistake in his manuscript that had just won a major international prize. The mistake was mathematical, but the consequences were profoundly physical, beginning the long road to a theory of chaos that simmered, without boiling, for nearly seventy years until computers became common lab equipment. Edward Lorenz of MIT, working on models of the atmosphere in the late 1960s, used one of the earliest scientific computers to expose the beauty and the complexity of chaotic systems. He discovered that the computer simulations were exponentially sensitive to the initial conditions, and the joke became that a butterfly flapping its wings in China could cause hurricanes in the Atlantic. In his computer simulations, Lorenz discovered what today is known as the Lorenz butterfly, an example of something called a “strange attractor”. But the term chaos is a bit of a misnomer, because chaos theory is primarily about finding what things are shared in common, or are invariant, among seemingly random-acting systems.

The third generation in concepts of motion, Space-Time, is indelibly linked with Einstein’s special theory of relativity, but Einstein was not its originator. Space-time was the brain child of the gifted but short-lived Prussian mathematician Hermann Minkowski, who had been attracted from Königsberg to the mathematical powerhouse at the University in Göttingen, Germany around the turn of the 20th Century by David Hilbert. Minkowski was an expert in invariant theory, and when Einstein published his special theory of relativity in 1905 to explain the Lorentz transformations, Minkowski recognized a subtle structure buried inside the theory. This structure was related to Riemann’s metric theory of geometry, but it had the radical feature that time appeared as one of the geometric dimensions. This was a drastic departure from all former theories of motion that had always separated space and time: trajectories had been points in space that traced out a continuous curve as a function of time. But in Minkowski’s mind, trajectories were invariant curves, and although their mathematical representation changed with changing point of view (relative motion of observers), the trajectories existed in a separate unchanging reality, not mere functions of time, but eternal. He called these trajectories world lines. They were static structures in a geometry that is today called Minkowski space. Einstein at first was highly antagonistic to this new view, but he relented, and later he so completely adopted space-time in his general theory that today Minkowski is almost forgotten, his echo heard softly in expressions of the Minkowski metric that is the background to Einstein’s warped geometry that bends light and captures errant space craft.

The fourth generation in the development of concepts of motion, Geometric Dynamics, began when an ambitious French physicist with delusions of grandeur, the historically ambiguous Pierre Louis Maupertuis, returned from a scientific boondoggle to Lapland where he measured the flatness of the Earth in defense of Newtonian physics over Cartesian. Skyrocketed to fame by the success of the expedition, he began his second act by proposing the Principle of Least Action, a principle by which all motion seeks to be most efficient by taking a geometric path that minimizes a physical quantity called action. In this principle, Maupertuis saw both a universal law that could explain all of physical motion, as well as a path for himself to gain eternal fame in the company of Galileo and Newton. Unfortunately, his high hopes were dashed through personal conceit and nasty intrigue, and most physicists today don’t even recognize his name. But the idea of least action struck a deep chord that reverberates throughout physics. It is the first and fundamental example of a minimum principle, of which there are many. For instance, minimum potential energy identifies points of system equilibrium, and paths of minimum distances are geodesic paths. In dynamics, minimization of the difference between potential and kinetic energies identifies the dynamical paths of trajectories, and minimization of distance through space-time warped by mass and energy density identifies the paths of falling objects.

Maupertuis’ fundamentally important idea was picked up by Euler and Lagrange, expanding it through the language of differential geometry. This was the language of Bernhard Riemann, a gifted and shy German mathematician whose mathematical language was adopted by physicists to describe motion as a geodesic, the shortest path like a great-circle route on the Earth, in an abstract dynamical space defined by kinetic energy and potentials. In this view, it is the geometry of the abstract dynamical space that imposes Galileo’s simple parabolic form on freely falling objects. Einstein took this viewpoint farther than any before him, showing how mass and energy warped space and how free objects near gravitating bodies move along geodesic curves defined by the shape of space. This brought trajectories to a new level of abstraction, as space itself became the cause of motion. Prior to general relativity, motion occurred in space. Afterwards, motion was caused by space. In this sense, gravity is not a force, but is like a path down which everything falls.

The fifth generation of concepts of motion, Quantum Coherence, increased abstraction yet again in the comprehension of trajectories, ushering in difficult concepts like wave-particle duality and quantum interference. Quantum interference underlies many of the counter-intuitive properties of quantum systems, including the possibility for quantum systems to be in two or more states at the same time, and for quantum computers to crack unbreakable codes. But this new perspective came with a cost, introducing fundamental uncertainties that are locked in a battle of trade-offs as one measurement becomes more certain and others becomes more uncertain.

Einstein distrusted Heisenberg’s uncertainty principle, not that he disagreed with its veracity, but he felt it was more a statement of ignorance than a statement of fundamental unknowability. In support of Einstein, Schrödinger devised a thought experiment that was meant to be a reduction to absurdity in which a cat is placed in a box with a vial of poison that would be broken if a quantum particle decays. The cruel fate of Schrödinger’s cat, who might or might not be poisoned, hinges on whether or not someone opens the lid and looks inside. Once the box is opened, there is one world in which the cat is alive and another world in which the cat is dead. These two worlds spring into existence when the box is opened—a bizarre state of affairs from the point of view of a pragmatist. This is where Richard Feynman jumped into the fray and redefined the idea of a trajectory in a radically new way by showing that a quantum trajectory is not a single path, like Galileo’s parabola, but the combined effect of the quantum particle taking all possible paths simultaneously. Feynman established this new view of quantum trajectories in his thesis dissertation under the direction of John Archibald Wheeler at Princeton. By adapting Maupertuis’ Principle of Least Action to quantum mechanics, Feynman showed how every particle takes every possible path—simultaneously—every path interfering in such as way that only the path with the most constructive interference is observed. In the quantum view, the deterministic trajectory of the cannon ball evaporates into a cloud of probable trajectories.

In our current complex times, the sixth generation in the evolution of concepts of motion explores Complex Systems. Lorenz’s Butterfly has more to it than butterflies, because Life is the greatest complex system of our experience and our existence. We are the end result of a cascade of self-organizing events that began half a billion years after Earth coalesced out of the nebula, leading to the emergence of consciousness only about 100,000 years ago—a fact that lets us sit here now and wonder about it all. That we are conscious is perhaps no accident. Once the first amino acids coagulated in a muddy pool, we have been marching steadily uphill, up a high mountain peak in a fitness landscape. Every advantage a species gained over its environment and over its competitors exerted a type of pressure on all the other species in the ecosystem that caused them to gain their own advantage.

The modern field of evolutionary dynamics spans a wide range of scales across a wide range of abstractions. It treats genes and mutations on DNA in much the same way it treats the slow drift of languages and the emergence of new dialects. It treats games and social interactions the same way it does the evolution of cancer. Evolutionary dynamics is the direct descendant of chaos theory that turned butterflies into hurricanes, but the topics it treats are special to us as evolved species, and as potential victims of disease. The theory has evolved its own visualizations, such as the branches in the tree of life and the high mountain tops in fitness landscapes separated by deep valleys. Evolutionary dynamics draws, in a fundamental way, on dynamic processes in high dimensions, without which it would be impossible to explain how something as complex as human beings could have arisen from random mutations.

These six generations in the development of dynamics are not likely to stop, and future generations may arise as physicists pursue the eternal quest for the truth behind the structure of reality.

Top 10 Books to Read on the History of Dynamics

Here are my picks for the top 10 books on the history of dynamics. These books have captivated me for years and have been an unending source of inspiration and information as I have pursued my own interests in the history of physics. The emphasis is on dynamics, rather than quantum and particle physics, although these traditional topics of “modern physics” have inherited many of the approaches of classical mechanics.

(1) Diacu, F. and P. Holmes (1996). Celestial encounters: The origins of chaos and stability. Princeton, N.J., Princeton Univ. Press.

Diacu and Holmes have written a clear, accessible and information-rich general history of the role that the solar system played in the development of dynamical theory, especially issues of the stability of the solar system.

(2) Pais, A. (2005) Subtle is the Lord: The Science and the Life of Albert Einstein: Oxford.

Pais has produced a masterpiece with his inside view of the historical development of Einstein’s ideas, for both special and general relativity. Through Pais’ story telling, it is possible to follow each turn in Einstein’s thinking as he proposed some of the most mind-bending ideas of physics.

(3) Thorne, K. S. (1994). Black holes and time warps : Einstein’s outrageous legacy. New York, W.W. Norton.

This book is an exuberant journey through the history of general relativity seen through the eyes of the recent Nobel Prize winner Kip Thorne. The book is full of details, many of them personal recollections as GR went from its early days through the “golden age” with John Wheeler located at the center of the motion.

(4) Schweber, S. S. (1994). QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga. Princeton, Princeton University Press.

Schweber has produced a master work in the same genre as Pais, describing the development of QED in such moment-by-moment detail that you feel you are living the history itself. The description of Feynman’s stumble into the world of the “grown ups” at the Shelter Island and Pocono Conferences is priceless.

(5) Bacaer, N. (2011). A Short History of Mathematical Population Dynamics, Springer.

This compact little book is one of my favorites in terms of conciseness and completeness. It tracks a history that is little known inside physics, but which has taken on out-sized importance in the new era of complex systems where evolutionary dynamics describes diverse systems from neural networks to genetic algorithms.

(6) Gleick, J. (1987). Chaos: Making a New Science, Viking.

Gleick’s book is an absolute classic. This was one of my first introductions into the history of modern physics when I read it at the end of my post-doc position at Bell Labs in 1989. It has been a role model for my own dive into the history of physics.

(7) Cassidy, David C. (2010). Beyond Uncertainty : Heisenberg, Quantum Physics, and The Bomb. New York, NY, Bellevue Literary Press.

Cassidy’s sequel to his first book on Heisenberg (Uncertainty) is in the same master genre as Pais and Schweber. Reading page by page allows you to live the history yourself as Heisenberg struggled to escape from an overbearing father (and a disastrous doctoral defense) to make his mark on the world of physics.

(8) Jammer, M. (1989), The conceptual development of quantum mechanics. Tomash Publishers Woodbury, N.Y., American Institute of Physics.

Although dry and a dense read, this book is definitive. If you ever want to understand step-by-step how quantum mechanics evolved from the early thinking of Bohr to the advanced transformations of Dirac and Jordan, this is the book you want as a reference. It is endlessly deep and detailed.

(9) Crowe, M. J. (2007), Mechanics from Aristotle to Einstein: Green Lion Press.

This book is filled with lots of myth-busting about the early days of physics. It’s amazing that what we call “Newtonian Physics” was mostly not invented by Newton himself, but by others … even by his nemesis Leibniz!

(10) Coopersmith, J. (2010), Energy, the Subtle Concept: The Discovery of Feynman’s Blocks from Leibniz to Einstein: Oxford, Oxford University Press.

Coopersmith shows how the history of concepts of work and energy is surprisingly obscure. Newton himself made no mention of energy, and it took nearly 100 years for a clear picture of energy to emerge, despite its central role in dynamical systems.


There are many wonderful review articles in review journals. A few of my favorites are:

Aubin A. and Dahan Dalmedico, D. (2002). “Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures”. Historia Mathematica, 29, 273-339.

Ginoux, J. M. and C. Letellier (2012). “Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept.” Chaos 22(2).

Gutzwiller, M. (1998), Moon-Earth-Sun: The oldest three-body problem, Reviews of Modern Physics, vol. 70, No. 2

Jenkins, A. (2013). “Self-oscillation.” Physics Reports-Review Section of Physics Letters 525(2): 167-222.

Morgan, G. J. (1998). “Emile Zuckerkandl, Linus Pauling, and the molecular evolutionary clock, 1959-1965.” Journal of the History of Biology 31(2): 155-178.



Galileo Unbound: The Physics and History of Dynamics

Welcome to Galileo Unbound: The History and Physics of Dynamics. This is the Blog site where you can find the historical background and the physical concepts behind many of the current trends in the physics of complex systems. It is written at the level of college undergraduates in fields of study like science or engineering. Advanced high school students should be able to find little gems here, too.

The topics here will fall under two headings that mirror my two recent books: Introduction to Modern Dyanamics (Oxford University Press, 2015) and Galileo Unbound (Oxford University Press, 2018). The first is a college junior-level physics textbook describing the mathematical details of modern dynamics. The second is a general interest book on the historical development of the same ideas. The physical concepts in both books will be expanded upon in this Blog at a general level of understanding. I hope you enjoy the broad range of topics that will appear here.

Good company in a journey makes the way seem shorter. — Izaak Walton