Solution for a free particle in a circular sector billiard (Q1011802)

Billiards problems, of considerable interest both theoretical and experimental, concern a particle confined to a closed region that rebounds elastically from the boundary walls and moves freely in between. The current paper fosuses on a billiards system in a circular sector. The Hamiltonian (in polar coordinates) for the motion inside a circular sector is \[ H= {1\over 2m}\Biggl(p^2_r+ {p^2_\varphi\over r^2}\Biggr)\quad\text{for }r< a, \] where \(a\) is the radius of the circle, \(m\) is the mass of the particle, and \(p_r\) and \(p_\varphi\) are components of momentum. There are two constants-of-the-motion here -- the energy and the angular momentum. Thus, the system is integrable. The authors describe the foliation of the phase space by invariant sets defined by constant energy, and then proceed to find every periodic or quasi-periodic orbit with a given winding number. The periodic orbits are of two types. The first, which is generic, is traversed only in one direction, and collisions with walls have different incoming and outgoing segments. The other kind of periodic is exceptional because it involves either orthogonal collision at a radia boundary, or a singular collision at the intersection of the radial boundary and the circular arc. In either case, the momenta \(p_r\) and \(p_\varphi\) are reversed (in sign) and the particle retraces its path in the opposite direction.