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The motion studied is that of a free particle sliding frictionlessly, namely, one having the Hamiltonian. where m is the mass of the particle, qi,i = 1,2 are the coordinates on the manifold, pi are the conjugate momenta: and. is the metric tensor on the manifold. — “Artin billiard - Wikipedia, the free encyclopedia”, en.wikipedia.org
Hadamard was able to show that every particle trajectory moves away from every other: that all trajectories have a positive Lyapunov exponent. The motion studied is that of a free particle sliding frictionlessly on the surface, namely, one having the Hamiltonian. — “Hadamard's dynamical system - Wikipedia, the free encyclopedia”, en.wikipedia.org
It consists of a stick that is free of the gravitational force, which can rotate frictionlessly in a plane around an axis located in one of its tips, and which is periodically kicked on the other tip. The variables θn and pn respectively determine. — “Standard map - Wikipedia, the free encyclopedia”, en.wikipedia.org