In chaos theory the separation of infinitesimally close trajectories has great importance. In present paper this behavior is investigated for classical magnetic billiard systems on Riemannian manifolds. The separation of the trajectories during the bounceless segments as well as at the reflections is studied generally, with a method similar to that of Jacobi fields for geodesic flows. For two-dimensional manifolds the results are also given in a natural coordinate frame, and they are illustrated in special (homogeneous) cases. We relate our issues to the known properties of the curvature of the horocycles, too.

The stability properties of the classical trajectories of charged particles are investigated in a two-dimensional inverse magnetic domain, where the magnetic field is zero inside the domain and constant outside. As an example, we present detailed analysis for stadium-shaped domain. In the case of infinite magnetic field, the dynamics of the system is the same as in the Bunimovich billiard, i.e., ergodic and mixing. However, for weaker magnetic fields, the phase space becomes mixed and the chaotic part gradually shrinks. The numerical measurements of the Lyapunov exponent (based on the technique of Jacobi fields) and the regular-to-chaotic phase space volume ratio show that both quantities can smoothly be tuned by varying the external magnetic field. A possible experimental realization of the inverse magnetic billiard is also discussed.

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