A sufficient condition for the existence of Hamiltonian bifurcations with continuous isotropy
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A SUFFICIENT CONDITION FOR THE EXISTENCE OF HAMILTONIAN BIFURCATIONS WITH CONTINUOUS ISOTROPY James Montaldi · Miguel Rodríguez-Olmos
Received: 16 October 2012 / Published online: 20 February 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract We present a framework for the study of the local qualitative dynamics of equivariant Hamiltonian flows specially designed for points in phase space with non-trivial isotropy. This is based on the classical construction of structure-preserving tubular neighborhoods for Hamiltonian Lie group actions on symplectic manifolds. This framework is applied to obtaining concrete and testable conditions guaranteeing the existence of bifurcations from symmetric branches of Hamiltonian relative equilibria. Keywords Hamiltonian bifurcations · Symmetry breaking · Momentum maps · Relative equilibria Mathematics Subject Classification 37C10 · 70H33 · 53D20
1 Introduction In this short note we outline a method for studying the qualitative dynamics of symmetric Hamiltonian systems near relative equilibria, with emphasis on those which lie in singular leaves of the momentum map, or equivalently, have continuous isotropy. Using this method we will state a sufficient, easily testable condition for the existence of a bifurcation from a continuous family of relative equilibria parametrized by momentum. For reasons of space this paper is of an expository nature and an announcement of some of the results contained in [5] where a more detailed analysis is carried out. Recall that a G-Hamiltonian system is a quintuple (P , ω, G, J, h), where (P , ω) is a smooth symplectic manifold, G is a Lie group acting on P by symplectomorphisms,
J. Montaldi School of Mathematics, University of Manchester, Manchester M13 9PL, UK e-mail: [email protected]
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M. Rodríguez-Olmos ( ) Department of Applied Mathematics IV, Technical University of Catalonia, Barcelona, Spain e-mail: [email protected]
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J. MONTALDI, M. RODRÍGUEZ-OLMOS
h ∈ C ∞ (P ) is a G-invariant Hamiltonian function and J : P → g∗ is an Ad∗ -equivariant momentum map satisfying ιξP ω = d J(·), ξ , ∀ξ ∈ g. Here g denotes the Lie algebra of G and ξP (z) = dtd |t=0 exp tξ · (z) is the evaluation at z of the fundamental vector field associated to the Lie algebra element ξ ∈ g. Throughout this paper we will assume that G is compact. The associated dynamical system on P is the flow of the Hamiltonian vector field defined implicitly by ιXh ω = dh. This flow is G-invariant, thus sending group orbits to group orbits for all times. A well known theorem by Noether states that J is constant along integral curves of Xh . A relative equilibrium is an integral curve of the dynamical system that is contained in a group orbit. They can be characterized (see [3]) as those points z ∈ P for which there exists an element ξ ∈ g such that Xh (z) = ξP (z).
(1)
The Lie algebra element ξ , which depends on z, is called a velocity for the relative equilibriu
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