Variational Principles for Lie-Poisson and Hamilton-Poincaré Equations

Abstract

As is well-known, there is a variational principle for theEuler–Poincar ́e equations on a Lie algebragof a Lie groupGobtainedby reducing Hamilton’s principle onGby the action ofGby, say, leftmultiplication. The purpose of this paper is to give a variational prin-ciple for the Lie–Poisson equations ong∗, the dual ofg, and also togeneralize this construction.The more general situation is that in which the original configura-tion space is not a Lie group, but rather a configuration manifoldQon which a Lie groupGacts freely and properly, so thatQ→Q/Gbecomes a principal bundle. Starting with a Lagrangian system onTQinvariant under the tangent lifted action ofG, the reduced equations on(TQ)/G, appropriately identified, are the Lagrange–Poincar ́e equations.Similarly, if we start with a Hamiltonian system onT∗Q, invariant un-der the cotangent lifted action ofG, the resulting reduced equations on(T∗Q)/Gare called the Hamilton–Poincar ́e equations.Amongst our new results, we derive a variational structure for theHamilton–Poincar ́e equations, give a formula for the Poisson structureon these reduced spaces that simplifies previous formulas of Montgomery,and give a new representation for the symplectic structure on the asso-ciated symplectic leaves. We illustrate the formalism with a simple, butinteresting example, that of a rigid body with internal rotors.