Billiards: A singular perturbation limit of smooth Hamiltonian flows

Nonlinear multi-dimensional Hamiltonian systems that are not near integrable typically have mixed phase space and a plethora of instabilities. Hence, it is difficult to analyze them, to visualize them, or even to interpret their numerical simulations. We survey an emerging methodology for analyzing a class of such systems: Hamiltonians with steep potentials that limit to billiards. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4722010]