We consider the effects of symmetry on the dynamics of a nonlinear hamiltonian system invariant under the action of a compact Lie group T, in the vicinity of an isolated equilibrium: in particular, the local existence and stability of periodic trajectories. The main existence result, an equivariant version of the Weinstein—Moser theorem, asserts the existence of periodic trajectories with certain prescribed symmetries Z c T x S
1
, independently of the precise nonlinearities. We then describe the constraints put on the Floquet operators of these periodic trajectories by the action of T. This description has three ingredients: an analysis of the linear symplectic maps that commute with a symplectic representation, a study of the momentum mapping and its relation to Floquet multipliers, and Krein Theory. We find that for some 2, which we call
cylospetral
, all eigenvalues of the Floquet operator are forced by the group action to lie on the unit circle; that is, the periodic trajectory is spectrally stable. Similar results for equilibria are described briefly. The results are applied to a number of simple examples such as T = SO(2), 0(2 ), Z
n
, D
n
, SU (2) ; and also to the irreducible symplectic actions of O(3) on spaces of complex spherical harmonics, modelling oscillations of a liquid drop.
Note to “Singular continuous Floquet operator for systems with increasing gaps”
