Variational Principles for Lie–poisson and Hamilton–poincaré Equations

As is well-known, there is a variational principle for the Euler–Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle for the Lie–Poisson equations on g∗, the dual of g, and also to generalize this construction. The more general situation is that in which the original configuration space is not a Lie group, but rather a configuration manifold Q on which a Lie group G acts freely and properly, so that Q → Q/G becomes a principal bundle. Starting with a Lagrangian system on TQ invariant under the tangent lifted action of G, the reduced equations on (TQ)/G, appropriately identified, are the Lagrange–Poincaré equations. Similarly, if we start with a Hamiltonian system on T ∗Q, invariant under the cotangent lifted action of G, the resulting reduced equations on (T ∗Q)/G are called the Hamilton–Poincaré equations. Amongst our new results, we derive a variational structure for the Hamilton–Poincaré equations, give a formula for the Poisson structure on these reduced spaces that simplifies previous formulas of Montgomery, and give a new representation for the symplectic structure on the associated symplectic leaves. We illustrate the formalism with a simple, but interesting example, that of a rigid body with internal rotors.