The Lagrange rigid body motion

Abstract

We discuss the motion of the three-dimensional rigid body about a fixed point under the influence of gravity, more specifically from the point of view of its symplectic structures and its constants of the motion. An obvious symmetry reduces the problem to a Hamiltonian flow on a four-dimensional submanifold of s o ( 3 ) × s o ( 3 ) ; they are the customary Euler-Poisson equations. This symplectic manifold can also be regarded as a coadjoint orbit of the Lie algebra of the semi-direct product group S O ( 3 ) × s o ( 3 ) with its natural symplectic structure. Finally the Lagrange motion is also a Hamiltonian flow on a coadjoint orbit in a kac-Moody Lie algebra; this approach has the virtue that the linearization in terms of elliptic integrals follows at once from a general theorem.