Integrable Systems and All That

This chapter is dedicated to the study of integrable mechanical systems and their properties. First we prove the Liouville theorem on systems which are integrable by quadratures. Then we introduce the action-angle canonical variables which are illustrated in several examples. We define the adiabatic processes and their invariant and prove that the action variables are adiabatic invariants when the frequency of the associated angle is non-zero. We briefly discuss the superintegrable systems and their geometric characterization. We introduce three modern techniques to construct/solve integrable systems: the projection method, the Lax pair representation, and the bi-Hamiltonian formulation. We conclude the chapter with a detailed analysis of two important “modern” integrable systems: the (non-periodic) Toda lattice and the Calogero–Moser system. Both models are shown to be integrable by a variety of different methods.