# Hadamard's dynamical system

In

physics andmathematics , the**Hadamard dynamical system**or**Hadamard's billiards**is a chaoticdynamical system , a type ofdynamical billiards . Introduced byJacques Hadamard in 1898 [*J. Hadamard, "Les surfaces à courbures opposées et leurs lignes géodésiques". "J. Math. Pures et Appl."*] , it is the first dynamical system to be proven chaotic.**4**(1898) pp. 27-73.The system considers the motion of a free (

friction less)particle on a surface of constant negativecurvature , the simplest compactRiemann 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 positiveLyapunov exponent .Frank Steiner [

*Frank Steiner, " [*] 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*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.Albert Einstein andErnst Mach .The system is particularly important in that in 1963,

Yakov Sinai , in studyingSinai's billiards as a model of the classical ensemble of aBoltzmann-Gibbs gas , was able to show that the motion of the atoms in the gas follow the trajectories in the Hadamard dynamical system.**Exposition**The motion studied is that of a free particle sliding frictionlessly on the surface, namely, one having the

Hamiltonian :$H(p,q)=frac\{1\}\{2m\}\; p\_i\; p\_j\; g^\{ij\}(q)$

where "m" is the mass of the particle, $q^i$, $i=1,2$ are the coordinates on the manifold, $p\_i$ are the

conjugate momenta ::$p\_i=mg\_\{ij\}\; frac\{dq^j\}\{dt\}$

and

:$ds^2\; =\; g\_\{ij\}(q)\; dq^i\; dq^j,$

is the

metric tensor on the manifold. Because this is the free-particle Hamiltonian, the solution to theHamilton-Jacobi equations of motion are simply given by thegeodesic s on the manifold.Hadamard was able to show that all geodesics are unstable, in that they all diverge exponentially from one-another, as $e^\{lambda\; t\}$ with positive

Lyapunov exponent :$lambda\; =\; sqrt\{frac\{2E\}\{mR^2$

with "E" the energy of a trajectory, and $K=-1/R^2$ being the constant negative curvature of the surface.

**References**

*Wikimedia Foundation.
2010.*

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