  We study periodic billiard trajectories on a compact Riemannian manifold with
boundary, by applying Morse theory to Lagrangian action functionals on the loop
space of the manifold. Based on the approximation method due to Benci-Giannoni,
we prove that nonvanishing of relative homology of a certain pair of loop
spaces implies the existence of a periodic billiard trajectory. We also prove a
parallel result for path spaces. We apply those results to show the existence
of short billiard trajectories and short geodesic loops. We also recover two
known results on the length of a shortest periodic billiard trajectory on a
convex body: Ghomi's inequality, and Brunn-Minkowski type inequality due to
Artstein-Ostrover.
