Introduced by Jacques Hadamard in 1898,[2] and studied by Martin Gutzwiller in the 1980s,[3][4] it is the first dynamical system to be proven chaotic.
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.
[5] 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.
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.
Because this is the free-particle Hamiltonian, the solution to the Hamilton–Jacobi equations of motion are 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