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The Tangled Tale of Phase Space David D The tangled tale of phase space David D. Nolte feature Phase space has been called one of the most powerful inventions of modern science. But its historical origins are clouded in a tangle of independent discovery and misattributions that persist today. David Nolte is a professor of physics at Purdue University in West Lafayette, Indiana. Figure 1. Phase space, a ubiquitous concept in physics, is espe- cially relevant in chaos and nonlinear dynamics. Trajectories in phase space are often plotted not in time but in space—as maps that show how trajectories intersect a region of phase space. Here, such a map is simulated by a so-called iterative Lozi map- ping, (x, y)→(1+y −∣x∣/2, −x). Each color represents the multiple intersections of a single trajectory starting from different initial conditions. Hamiltonian Mechanics is geometry in phase space. ern physics (figure 1). The historical origins have been further —Vladimir I. Arnold (1978) obscured by overly generous attribution. In virtually every textbook on dynamics, classical or statistical, the first refer- Listen to a gathering of scientists in a hallway or a ence to phase space is placed firmly in the hands of the French coffee house, and you are certain to hear someone mention mathematician Joseph Liouville, usually with a citation of the phase space. Walk down the science aisle of the local book- 1838 paper in which he supposedly derived the theorem on store, and you will surely catch a glimpse of a portrait of a the conservation of volume in phase space.2 (The box on page strange attractor, the powerful visual icon of phase space. 34 gives a modern derivation.) In fact, in his paper Liouville Though it was used originally to describe specific types of makes no mention of phase space, let alone dynamical sys- dynamical systems, today “phase space” has become syn- tems. Liouville’s paper is purely mathematical, on the behav- onymous with the idea of a large parameter set: Whether they ior of a class of solutions to a specific kind of differential are stock prices in economics, the dust motes in Saturn’s equation. Though he lived for another 44 years, he was ap- rings, or high-energy particles in an accelerator, the degrees parently unaware of his work’s application to statistical me- of freedom are loosely called the phase space of the respective chanics by others3 even within his lifetime. Therefore, Liou- systems. The concept and its name are embedded in our sci- ville’s famous paper, cited routinely by all the conventional entific fluency and cultural literacy. In his popular book Chaos textbooks, and even by noted chroniclers of the history of on the history and science of chaos theory, James Gleick calls mathematics, as the origin of phase space, surprisingly is not! phase space “one of the most powerful inventions of modern How did we lose track of the discovery of one of our science.”1 But who invented it? Who named it? And why? most important modern concepts in physics? If it was not dis- The origins of both the concept of phase space and its covered by Liouville, then by whom and when and why? name are historically obscure—which is surprising in view And where did it get its somewhat strange name of “phase” of the central role it plays in practically every aspect of mod- space? Where’s the phase? © 2010 American Institute of Physics, S-0031-9228-1004-010-X April 2010 Physics Today 33 Hamiltonian flow in phase space Hamilton’s dynamical equations connect position x with Volume in phase space is conserved under Hamiltonian flow, a momentum px through the Hamiltonian H: property known today as Liouville’s theorem. dx ∂Hxp(, ) dp ∂Hxp(, ) = x , x =− x . dt ∂px dt ∂x px Dynamics can generally be described as a flow or trajectory Phase given by a vector differential equation, point Trajectory ηη˙ =().f In phase space, η represents the position or phase point, and the flow is expressed through Hamilton’s equations. In two- dV = dpx dx Δp 1 dimensional phase space, x Δx ∂H ∂H x η =(xp , ), f()=η ,− . x ( ( ∂px ∂x dV2 = dV1 The evolution of a volume element dV = dpxdx in phase space is given by 1 dV ∂ ∂H ∂ ∂H =·=∇f ((+−(( =0 Vdt ∂x ∂px ∂px ∂x The search for the origins of phase space presents two satisfies the equation challenges. The first is to identify those people who con- du n ∂P = ∑ i u. tributed to the development of the concept of phase space. To ( ∂x ( do that, we will start back at the time of Liouville to find out dt i =1 i what role he really did play in the story and how his contri- If the expression in parentheses is zero, then u is a constant. bution found its way into modern textbooks. The second Furthermore, if the arbitrary constants ai are chosen to be val- challenge is to discover who gave phase space its fully mod- ues of xi at time t = 0, then the system has the solution ern name. To answer that question, we will be led to a now ∂x ()t ut()=deti =1 obscure encyclopedia article published in 1911 that had ety- (∂x (0) ( mological side effects not fully intended by its authors. j Liouville’s theorem because u(0) is clearly equal to 1. Liouville’s 1838 paper appeared only a few years after Liouville (1809–82) was perhaps the most renowned French William Rowan Hamilton (1805–65) published his dynamics4 mathematician of the mid-19th century. That era was the in 1834 and 1835, yet Liouville made no reference to his the- golden age of differential calculus and the beginning of dif- orem’s application to dynamics. ferential geometry. Liouville displayed a virtuoso breadth of The connection with mechanics was made in 1842 by the expertise in topics ranging from number theory and complex Prussian mathematician Carl Gustav Jacob Jacobi (1804–51), analysis to differential geometry and topology. He is known who recognized that the differential equations that Liouville among mathematicians and physicists for several mathemat- had studied could describe mechanical systems5 (figure 3). In ical theorems, most notably the Sturm–Liouville theory of in- Hamilton’s dynamics, position coordinates xi and momen- tegral equations. The motivation for much of his work came tum coordinates pi evolve according to the Hamiltonian func- from physical problems in celestial mechanics, but he also tion H: drew from electrodynamics and the theory of heat. The prop- dx ∂Hdp ∂H erties and solutions of differential equations of many vari- i ==−., i ables were among his main areas of interest, and in that con- dt∂pi dt ∂xi text he was working on the solution of differential equations with constant integrals in the late 1830s. To apply Liouville’s theorem, Jacobi made the assignments In modern notation, the original formulation of Liou- (again in modern notation) ville’s theorem2 (figure 2) states that given a system of n first- xxxin= 1:=, in = 1: inn = +1:2 =; p in = 1: order differential equations dx ∂H ∂H i Pin=1: ==.,−Pin= +1:2 n =Ptxin (,12 , x ,..., x ), ∂p ∂x dt i in=1: i in=1: if a complete set of solutions is Therefore, xxtaa= (, , ,..., a ), ii12 n 2nn∂P ∂2H n∂2H ∑∑i =−=0, ∑ ∂ ∂ where the ai are arbitrary constants, then the Jacobian deter- i =1∂xi i =1∂xiip i =1 ∂piix minant ∂x and, according to Liouville’s observation, the Jacobian deter- ut( ) = det i (∂a ( minant u is constant. j Explicitly referencing Liouville’s 1838 paper in the 1866 34 April 2010 Physics Today www.physicstoday.org multidimensional space. But in the 1700s it was not natural. Variables were alge- braic entities, not coordinate axes. And while the solutions to equations of multi- ple variables could lead to loci of points with multiple indices, there was no for- mal thought that they represented geo- metric objects in higher dimensions. Amazingly, the surface areas and vol- umes of spheres in four dimensions (and higher) had been derived by Jacobi as early as 1834, but to him they were simply integrals over products of differentials. All that would change rather sud- denly in the 1840s as Plücker in Germany and Cayley and James Joseph Sylvester in the UK parameterized projective geometry and found extensions beyond the ordinary three dimensions of our Figure 2. Liouville’s theorem, as derived in Joseph Liouville’s paper of 1838 tangible world. Cayley, in his 1843 paper entitled “Note Sur la Théorie de la Variation des constantes arbitraires” (“Note 2 titled “Chapters in the Analytical Geom- on the Theory of the Variation of Arbitrary Constants”), often cited as the etry of (n) Dimensions,” was the first to original proof of the conservation of phase space. Liouville never applied the take the bold step of referring to a geom- theorem to mechanics—to him, it dealt just with a specific kind of differential etry of more than three dimensions. equation. After that, the stage was set for the “in- vention” of multiple dimensions when Grassmann developed the concept of an n-dimensional vector space in 1844. The publication of his lectures of 1842–43, Jacobi was the first to culmination of the multidimensional trend in analytic geom- put Liouville’s mathematical theorem into a mechanical con- etry came with Riemann’s lecture on the foundations of text. What Jacobi did not do, and indeed could not do in his geometry, delivered at the University of Göttingen in 1854, in time, was to represent mechanical systems within a general- which he systematized concepts of curved spaces that were ized space.
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