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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
and mathematics, the Hadamard dynamical system (also called Hadamard's billiard or the Hadamard–Gutzwiller model) is a
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids ...
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
, a type of
dynamical billiards A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of spe ...
. Introduced by
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a tea ...
in 1898, and studied by
Martin Gutzwiller Martin may refer to: Places * Martin City (disambiguation) * Martin County (disambiguation) * Martin Township (disambiguation) Antarctica * Martin Peninsula, Marie Byrd Land * Port Martin, Adelie Land * Point Martin, South Orkney Islands Austr ...
in the 1980s, it is the first dynamical system to be proven
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids ...
. The system considers the motion of a free (
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
less)
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, fro ...
on the
Bolza surface In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL_2(3) of order 48 ...
, i.e, a two-dimensional surface of genus two (a donut with two holes) and constant negative
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
; this is a compact
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
. Hadamard was able to show that every particle trajectory moves away from every other: that all trajectories have a positive
Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with in ...
. 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
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
and
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mac ...
. The system is particularly important in that in 1963,
Yakov Sinai Yakov Grigorevich Sinai (russian: link=no, Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a Russian-American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dy ...
, 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.


Exposition

The motion studied is that of a free particle sliding frictionlessly on the surface, namely, one having the Hamiltonian :H(p,q)=\frac p_i p_j g^(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 In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
: :p_i=mg_ \frac and :ds^2 = g_(q) dq^i dq^j\, is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allo ...
on the manifold. Because this is the free-particle Hamiltonian, the solution to the Hamilton–Jacobi equations of motion are simply given by the
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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^ with positive
Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with in ...
:\lambda = \sqrt with ''E'' the energy of a trajectory, and K=-1/R^2 being the constant negative curvature of the surface.


References

{{Reflist Chaotic maps Ergodic theory