Abstract: These notes were written for an audience of graduate students or upper-level undergraduates. Some experience with nonlinear ordinary differential equations and some familiarity with classical Hamiltonian systems is assumed. These notes are centred around the problem of perturbed integrable Hamiltonian systems, also referred to as nearly integrable systems. This topic has a long history, but the 'modern' theory begins with Poincare's work in 1900. An important milestone was Kolmogorov, Arnold and Moser's Theorem (KAM Theorem) in the 1960's. This theorem addresses what happens to the stable quasi-periodic motions of the integrable system under small perturbations. Another milestone was Nekhoroshev's Theorem in 1977. This theorem addresses the behaviour of all the solutions of the perturbed system. Whereas the KAM theorem discusses the perpetual stability of solutions, Nekhoroshev's theorem discusses the effective stability of solutions of integrable systems under perturbation. During the proof of Nekhoroshev's theorem we also learn what the mechanism is that allows the perturbation to destroy the stable motions of the unperturbed system.
The notes begin with a general discussion of dynamical systems and the notions of stability and invariant sets. Then we specialize to Hamiltonian ordinary differential equations and discuss action-angle variables and their significance. We state and discuss the KAM theorem. The last, and most lengthy, section of the notes discusses the proof of Nekhoroshev's theorem. The notes conclude with a list of references (articles and books).