Hamiltonian Systems Near Relative Periodic Orbits

Abstract

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, 70H33PII. S11111111013877601. 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 [25,39,43] and can therefore be found in virtually any physical application with a continuous symmetry group. Specific examples for which relative periodic orbits have been discussed or could be found by applying the Weinstein-Moser theorem to stable relative equilibria include rigid bodies [1,31,28,24], deformable bodies [8,27,13], gravitational N-body problems [32,47], molecules [17,19,20,34,48], and point vortices [26,46,38].Existing theoretical work on Hamiltonian relative periodic orbits includes results on their stability [41,42] and on their persistence to nearby energy-momentum levels in the case of compact symmetry groups [33]. However, stability, persistence, and bifurcations are still a long way from being well understood, especially in the presence of actions of noncompact symmetry groups with nontrivial isotropy subgroups. Our main aim with this paper is to

Numerical Continuation of Hamiltonian Relative Periodic Orbits

“…The reversing spatio-temporal symmetry L ρ of the Figure Eight is then isomorphic to L ρ D 6 Z 2 . Reversing symmetries ρ act on momenta µ ∈ g * and infinitesimal symmetries ξ ∈ g as follows (see [37]): ρµ = −(Ad * ρ ) −1 µ, ρξ = −Ad ρ ξ. Theorem 3.17 and Corollary 4.3 can be extended to include time-reversing symmetries by just replacing the spatio-temporal symmetry groups of the original and bifurcating solutions with the corresponding time-reversal spatio-temporal symmetry groups. One can then check that the type I rotating Eight at angular momentum 0 has the reversing symmetry ρ I = ρκ 2 (132) with respect tox, that the type II rotating Eight has the reversing symmetry ρ II = ρκ 2 κ 3 (132) and that the type III rotating Eight has the reversing symmetry ρ III = ρ I .…”

“…Specific examples where relative periodic orbits have been discussed or could be found near relative equilibria include gravitational N -body problems, molecules, underwater vehicles, vortices in ideal fluids and continuum mechanics, see e.g. [4,25,26,28,34,37] and the references therein. A relative equilibrium is an equilibrium after symmetry reduction, and a relative periodic orbit (RPO) is a trajectory which is periodic after symmetry reduction.…”

Classical Dynamics of Fullerenes

“…For any affine mapping φ ∈ Aff(N, M ), an analogous representation is valid when the affine coordinates are fixed (i.e., origins of the reference frames and the linear bases) [7,10,19,20,[23][24][25][26][27][29][30][31][32][33][34][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][55][56][57][58]60]. Let us also quote the following 'representation' property of the operation L:…”

“…The manifolds of the affine isomorphisms of N onto M will be denoted by AffI(N, M Affine structures of N , M are sufficient for defining affine degrees of freedom of extended bodies or of their structural elements [58]. Situation is more complicated in the case of dynamics.…”

“…It is also clear that the i-th component of the Euler coordinates of the center of mass is given by 58) and therefore one has also…”