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| In [[physics]] and [[mathematics]], the '''Hadamard dynamical system''' or '''Hadamard's billiard''' is a [[chaos theory|chaotic]] [[dynamical system]], a type of [[dynamical billiards]]. Introduced by [[Jacques Hadamard]] in 1898,<ref>{{Cite journal |first=J. |last=Hadamard |title={{lang|fr|Les surfaces à courbures opposées et leurs lignes géodésiques}} |journal=J. Math. Pures et Appl. |volume=4 |year=1898 |issue= |pages=27–73 |doi= }}</ref> it is the first dynamical system to be proven [[Randomness|chaotic]].
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| The system considers the motion of a free ([[friction]]less) [[Subatomic particle|particle]] on 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]].
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| 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.<ref>{{Cite book |first=Frank |last=Steiner |chapter=Quantum Chaos |title={{lang|de|Schlaglichter der Forschung: Zum 75. Jahrestag der Universität Hamburg 1994}} |editor-first=R. |editor-last=Ansorge |year=1994 |location=Berlin |publisher=Reimer |pages=542–564 |isbn=3-496-02540-9 |arxiv=chao-dyn/9402001 }}</ref> He points out that the study was widely disseminated, and considers the impact of the ideas on the thinking of [[Albert Einstein]] and [[Ernst Mach]].
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| The system is particularly important in that in 1963, [[Yakov Sinai]], in studying [[Sinai's billiards]] as 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.
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| ==Exposition==
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| The motion studied is that of a free particle sliding frictionlessly on the surface, namely, one having the [[Hamiltonian (quantum mechanics)|Hamiltonian]]
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| :<math>H(p,q)=\frac{1}{2m} p_i p_j g^{ij}(q)</math> | |
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| where ''m'' is the mass of the particle, <math>q^i</math>, <math>i=1,2</math> are the coordinates on the manifold, <math>p_i</math> are the [[conjugate momenta]]:
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| :<math>p_i=mg_{ij} \frac{dq^j}{dt}</math>
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| and
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| :<math>ds^2 = g_{ij}(q) dq^i dq^j\,</math>
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| is the [[metric tensor]] on the manifold. Because this is the free-particle Hamiltonian, the solution to the [[Hamilton–Jacobi equation|Hamilton–Jacobi equations of motion]] are simply given by the [[geodesic]]s on the manifold.
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| Hadamard was able to show that all geodesics are unstable, in that they all diverge exponentially from one another, as <math>e^{\lambda t}</math> with positive [[Lyapunov exponent]]
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| :<math>\lambda = \sqrt{\frac{2E}{mR^2}}</math>
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| with ''E'' the energy of a trajectory, and <math>K=-1/R^2</math> being the constant negative curvature of the surface.
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| ==References==
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| {{Reflist}}
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| [[Category:Chaotic maps]]
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| [[Category:Ergodic theory]]
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