Degenerate billiards

Abstract

In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Then collisions are rare. We study trajectories of degenerate billiards which have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an antiintegrable limit.

Degenerate billiards in celestial mechanics

“…Before formulating the main results we need to recall some definitions from [7], see also [10]. The Hamiltonian is constant along collision chains of a degenerate billiard, so let us fix energy H = E. The restriction of the Hamiltonian system (M, H) to the energy level will be denoted (M, H = E).…”

“…Consider the billiard system (Ω ε , Σ ε , H) in the domain Ω ε = M \ N ε with the boundary ∂Ω ε = Σ ε and Hamiltonian H. As ε → 0, it approaches the degenerate billiard (M, N, H) with the scatterer N . In [7] it is proved that for small ε > 0 nondegenerate collision chains of this degenerate billiard are shadowed by trajectories of the billiard system in Ω ε . For a discrete set N , this was shown earlier in [14], see also [12].…”

“…In particular, for any x 0 ∈ D E there is r > 0 such that a pair of points in the ball B r (x 0 ) is joined by a geodesic in B r (x 0 ). 7 We identify curves which differ by an orientation preserving reparametrization.…”

Counting Collisions in an N-Billiard System Using Angles Between Collision Subspaces

“…This is exactly the case for N masses on a line. Similar degenerate billiard constructions have been studied [4,5] along with their connections to celestial mechanics.…”