Non-integrability of n-centre billiards

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.