Non-integrability of n-centre billiards

TL;DR

This paper studies a class of planar billiard systems influenced by multiple centers, demonstrating that with at least two centers and a distant wall, the system is inherently non-integrable and chaotic, extending classical results.

Contribution

It establishes conditions under which n-centre billiards are non-integrable, including a geometric criterion for chaos in the two-centre case, advancing understanding of billiard dynamics.

Findings

Systems with two or more centers and distant walls are non-integrable and chaotic.

A geometric condition guarantees chaos in the classical two-centre problem.

The study extends classical billiard models to more complex potential-influenced systems.

Abstract

We investigate a class of mechanical billiards, where a particle moves in a planar region under the influence of an n-centre potential and reflects elastically on a straight wall. Motivated by Boltzmann's original billiard model we explore when such systems fail to be integrable. We show that for any positive energy, if there are at least two centres and the wall is placed sufficiently far away, the billiard dynamics is necessarily non-integrable and exhibits chaotic behaviour. In the classical Newtonian two-centre problem, we identify a simple geometric condition on the wall that likewise guarantees chaos, even though the free two-centre system is integrable.