Integrable Systems on Poisson Manifolds

Abstract

In this chapter we give the basic definitions of Liouville integrable systems on Poisson manifolds, we prove some key propositions and we give simple examples to illustrate the theory. While the definition of Liouville integrability is given on a general Poisson manifold, we will restrict ourselves to real Poisson manifolds in Section 4.3, where we will discuss the classical Liouville Theorem and the Action-Angle Theorem, which are, as such, only valid in the real case. For a complex version of the Liouville Theorem, we refer to Section 6.3. Lax equations, which often represent a vector field of an integrable system, are the subject of Sections 4.4 and 4.5.