Periodic solutions near equilibria of symmetric Hamiltonian systems

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 S1, 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 ), Zn, Dn, SU (2) ; and also to the irreducible symplectic actions of O(3) on spaces of complex spherical harmonics, modelling oscillations of a liquid drop.

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