Linear stability in billiards with potential

A general formula for the linearized Poincare map of a billiard with a potential is derived. Arc length and parallel component of the velocity are shown to be canonical coordinates for the map from bounce to bounce. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. Four billiards with potentials for which the free motion is integrable are treated as examples, the linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three-body problem. The linear stability of periodic orbits with periods one and two is analysed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.