magnetically confined rigid bodies

We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincaré section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry properties of the Hamiltonian system is described and the effects of time reversing symmetries are included. Our analysis yields new results on the stability and persistence of Hamiltonian relative periodic orbits and provides the foundations for a bifurcation theory. The results are applied to a finite dimensional model for the dynamics of a deformable body in an ideal irrotational fluid.

Abstract. We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincaré section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry properties of the Hamiltonian system is described, and the effects of time reversing symmetries are included. Our analysis yields new results on the stability and persistence of Hamiltonian relative periodic orbits and provides the foundations for a bifurcation theory. The results are applied to a finite dimensional model for the dynamics of a deformable body in an ideal irrotational fluid. Key words. relative periodic orbits, equivariant Hamiltonian systems, noncompact groups AMS subject classifications. 37J15, 37J20, 53D20, 70H33 PII. S1111111101387760 1. Introduction. Relative periodic orbits are periodic solutions of a flow induced by an equivariant vector field on a space of group orbits. In applications they typically appear as oscillations of a system which are periodic when viewed in some rotating or translating frame. They therefore generalize relative equilibria, for which the "shape" of the system remains constant in an appropriate frame. Relative periodic orbits are ubiquitous in Hamiltonian systems with symmetry. For example, generalizations of the Weinstein-Moser theorem show that they are typically present near stable relative equilibria Existing theoretical work on Hamiltonian relative periodic orbits includes results on their stability

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equilibria