Galileo was proud of his parabolic trajectory. In his first years after arriving at the university in Padua, he had worked with marked intensity to understand the mathematical structure of the trajectory, arriving at a definitive understanding of it by 1610—just as he was distracted by his friend Paolo Sarpi who suggested he improve on the crude Dutch telescopes starting to circulate around Venice. This distraction lasted 20 years, only to end under his house arrest, when he finally had time to complete and publish his work on motion in his Two New Sciences.
Galileo’s trajectory was a solitary arc with specific initial conditions. Similarly, Kepler’s orbits were solitary ellipses, one for each of the planets. When Newton posited his second axiom, in Latin prose (no equation) outlining the differential equation , he integrated it to obtain its integral curve, once again focusing on special cases defined by specific initial conditions. Even Lagrange, freed from Newton’s law, was still bound in his use of the principle of least action to solitary beginnings and endings, to the single path between two points that minimized the mechanical action. This legacy of great figures in the history of physics—Galileo, Kepler, Newton, Lagrange—along with a legion of introductory physics teachers today, helps to perpetuate the perspective that physics treats solitary trajectories.
Yet over a hundred years ago, Henri Poincaré took a decisive step away from the solitary trajectory and began thinking about dynamical systems as geometric problems in dynamical spaces. He was not the first—the nineteenth century mathematicians Rudolph Lipschitz and Gaston Darboux, inspired by Bernhard Riemann, had already begun to think of dynamics in terms of geodesics in metric spaces—but Poincaré was the first to create a new toolbox of techniques that treated dynamical systems as if they were the flow of fluids. He took a fundamental step away from concentrating on single trajectories one at a time, to thinking in terms of the set of all trajectories, treated simultaneously as a field of flow lines in dynamical spaces.
This was the dawn of phase space. Phase space is a multidimensional hyperspace that is like a bucket holding all possible trajectories of a complex system. James Gleick, the noted chronicler of science, has called the invention of phase space “one of the most important discoveries of modern science”. Phase space was invented in fits and starts (and at times independently) across a span of 70 years, beginning with Liouville in 1838, then added to by Jacobi, Boltzmann, Maxwell, Poincaré, Gibbs and finally Ehrenfest in 1911. A phase-space diagram, also known as a phase-space “portrait,” consists of a set of flow lines that swirl around special points called fixed points that act as sources or sinks or centers, and the flow is constrained by special lines called nullclines and separatrixes that corral the flow lines into regions called basins. These complex flows are like string art, and phase-space portraits like Lorenz’s Butterfly have become cultural memes.
Attached to each flow line on a phase portrait is a tangent vector, constituting a vector field that fills phase space. These tangent vectors are to dynamics what the vector fields of electric and magnetic phenomena are to electromagnetics. A broad class of complex systems with their flows can be described through the deceptively simple mathematical expression for the vector fieldwhere the “dot” is the time derivative, and the variables and the function are vector quantities. The vector is the set of all positions and all momenta for a system with N degrees of freedom. When the physical system is composed of interacting massive particles that conserve energy (like the atoms in an ideal gas or planets in orbit), then the flow equation reduces to Hamilton’s equations of dynamics. Yet dynamical systems treated by flows are much more general, capable of capturing the behavior of firing neurons; the beating heart,; economic systems like world trade; the evolution of species in ecosystems; the orbits of photons around black holes. These have all become the objects of study for modern dynamics. It would not be an exaggeration to say that is to modern dynamics what was to classical dynamics.
The flows of physics encompass such a broad sweep of subjects, that it is sometimes difficult to see the common thread that connects them all. Yet, the phase-space phase portrait, and computer techniques for the hypervisualization of many dimensions are becoming common tools, providing a unifying language with which to explain and predict the behavior of systems that were once too complex to address. By learning this language, a new generation of scientists will cut across the Babel of so many disciplines.
Featured image credit: The International Space Station by Paul Larkin. Public Domain via Unsplash.