Variational Principles for Lie-Poisson and Hamilton-Poincaré Equations

Cendra, H., J. E. Marsden, S. Pekarsky and T. S. Ratiu

Moscow Mathematical Journal 3, (2003) 833-867

Abstract:

As is well-known, there is a variational principle for the Euler-Poincaré equations on a Lie algebra $\mathfrak{g}$ 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 $\mathfrak{g}^{\ast}$, the dual of $\mathfrak{g}$, 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 $\rightarrow$ 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.