Hadamard's dynamical system
physicsand mathematics, the Hadamard dynamical system or Hadamard's billiards is a chaotic dynamical system, a type of dynamical billiards. Introduced by Jacques Hadamardin 1898 [J. Hadamard, "Les surfaces à courbures opposées et leurs lignes géodésiques". "J. Math. Pures et Appl." 4 (1898) pp. 27-73.] , it is the first dynamical system to be proven chaotic.
The system considers the motion of a free (
frictionless) particleon a surface of constant negative curvature, the simplest compact Riemann surface, which is the surface of genus two: a donut with two holes. Hadamard was able to show that every particle trajectory moves away from every other: that all trajectories have a positive Lyapunov exponent.
Frank Steiner [Frank Steiner, " [http://arxiv.org/pdf/chao-dyn/9402001 Quantum Chaos] ", "Fetschrift Universitãt Hamburg 1994: Schlaglichter der Forschung zum 75. Jahrslag", Ed. R. Ausorge, (1994) pp. 542-564.] argues that Hadamard's study should be considered to be the first-ever examination of a chaotic dynamical system, and that Hadamard should be considered the first discoverer of chaos. He points out that the study was widely disseminated, and considers the impact of the ideas on the thinking of
Albert Einsteinand Ernst Mach.
The system is particularly important in that in 1963,
Yakov Sinai, in studying Sinai's billiardsas a model of the classical ensemble of a Boltzmann-Gibbs gas, was able to show that the motion of the atoms in the gas follow the trajectories in the Hadamard dynamical system.
The motion studied is that of a free particle sliding frictionlessly on the surface, namely, one having the
where "m" is the mass of the particle, , are the coordinates on the manifold, are the
metric tensoron the manifold. Because this is the free-particle Hamiltonian, the solution to the Hamilton-Jacobi equations of motionare simply given by the geodesics on the manifold.
Hadamard was able to show that all geodesics are unstable, in that they all diverge exponentially from one-another, as with positive
with "E" the energy of a trajectory, and being the constant negative curvature of the surface.
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