Introduction

The purpose of these notes is to give a brief survey of bifurcation theory of Hamiltonian systems with symmetry; they are a slightly extended version of the five lectures given by JM on Hamiltonian Bifurcations with Symmetry. We focus our attention on bifurcation theory near equilibrium solutions and relative equilibria. The notes are composed of two parts. In the first, we review results on nonlinear normal modes in equivariant Hamiltonian systems, generic movement of eigenvalues in equivariant Hamiltonian matrices, one and two parameter bifurcation of equilibria and the Hamiltonian-Hopf Theorems with symmetry. The second part is about local dynamics near relative equilibria. Particular topics discussed are the existence, stability and persistence of relative equilibria, bifurcations from zero momentum relative equilibria and examples.

We begin with some basic facts on Lie group actions on symplectic manifolds and Hamiltonian systems with symmetry. The reader should refer to Ratiu's lectures for more details and examples.

Semisymplectic actions A Lie group G acts semisymplectically on a symplectic manifold (P, ω) if g*ω = ±ω. In this case the choice of sign determines a homomorphism X : G → Z2 called the temporal character, such that g*ω = X(g)ω. We denote the kernel of X by G+; it consists of those elements acting symplectically, and if G does contain antisymplectic elements then G+ is a subgroup of G of index 2. Some details on semisymplectic actions can be found in [MR00].