- 05-30 P. Balint and I. P. Toth
- Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards
Jan 21, 05
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Abstract. When considering hyperbolicity in multi-dimensional Hamiltonian sytems,
especially in higher dimensional billiards, the literature usually
distinguishes between dispersing and defocusing mechanisms. In this paper
we give a unified treatment of these two phenomena, which also covers the
important case when the two mechanisms mix. Two theorems on the hyperbolicity
(i.e. non-vanishing of the Lyapunov exponents) are proven that are hoped to
be applicable to a variety of situations.
As an application we investigate soft billiards, that is, replace the hard
core collision in dispersing billiards with disjoint spherical scatterers
by motion in some spherically symmetric potential. Analogous systems in two
dimensions have been widely investigated in the literature, however, we are
not aware of any mathematical result in this multi-dimensional case.
Hyperbolicity is proven under suitable conditions on the potential. This
way we give a natural generalization of the hyperbolicity results
obtained before in two dimensions for a large class of potentials.