The system considers the motion of a free (frictionless) 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.
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 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.
where m is the mass of the particle, , are the coordinates on the manifold, are the conjugate momenta:
Hadamard was able to show that all geodesics are unstable, in that they all diverge exponentially from one-another, as with positive Lyapunov exponent
with E the energy of a trajectory, and being the constant negative curvature of the surface.