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 $\mathfrak g$ 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 $\mathfrak g^*$, 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 $Q\to Q/G$ 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.