Integrability

Abstract

AbstractIn this chapter, we study the concept of integrability of Hamiltonian systems in a systematic way. We start with the very notion of an integrable system and with a number of examples: the two-body problem, the two-centre problem, the top, the spherical pendulum and the Toda lattice. Thereafter, we analyse Lax pairs in the context of Hamiltonian systems on coadjoint orbits. In particular, we show that the Toda lattice can be understood in this framework. Next, we study the local geometric structure of integrable systems. We prove the Arnold Theorem, discuss the relation with symplectic reduction and present the construction of local action and angle variables in detail. As an application, we construct action and angle variables for a number of examples. We also show that action and angle variables are well adapted to the study of small perturbations of integrable systems. Thereafter, we give an introduction to global aspects in the spirit of Nekhoroshev and Duistermaat, with some emphasis on monodromy. This phenomenon will be illustrated in detail for the case of the spherical pendulum. Finally, we present a generalization to the concept of noncommutative integrability in the sense of Mishchenko and Fomenko. We prove both the Nekhoroshev Theorem and the Mishchenko-Fomenko Theorem and illustrate the latter for the case of the Euler top.KeywordsAngle VariableParallel TransportInvariant TorusToda LatticeCoadjoint OrbitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.