Symmetries of Hamiltonian systems on symplectic and Poisson manifolds

Abstract

This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions in differential geometry. Full proofs of many results are provided. The Marsden-Weinstein reduction procedure and the Euler-Poincaré equation are discussed. Symplectic cocycles involved when a momentum map is equivariant with respect to an affine action whose linear part is the coadjoint action are presented. The Hamiltonian actions of a Lie group on its cotangent bundle obtained by lifting the actions of the group on itself by translations on the left and on the right are fully discussed. We prove that there exists a two-parameter family of deformations of these actions into a pair of mutually symplectically orthogonal Hamiltonian actions whose momentum maps are equivariant with respect to an affine action involving any given Lie group symplectic cocycle. In the last section three classical examples are considered: the spherical pendulum, the motion of a rigid body around a fixed point and the Kepler problem. For each example the Euler-Poincaré equation is derived, the first integrals linked to symmetries are given. In this Section, the classical concepts of vector calculus on an Euclidean three-dimensional vector space (scalar, vector and mixed products) are used and their interpretation in terms of concepts such as the adjoint or coadjoint action of the group of rotations are explained.