Lie-Poisson structure for the homogeneous motion of self-gravitating compressible fluids

In this paper we apply the theory developed by Marsden, Ratiu, and Weinstei n for the reduction of a Hamiltonian system defined on the cotangent bundle of a Lie group to a Hamiltonian system in the coalgebra of a semidirect pr oduct to study the motion of a self-gravitating homogeneous compressible id eal fluid with a variable ellipsoidal boundary, assuming that the motions a re given by invertible linear transformations. The relation between the Lie -Poisson equations obtained and the classical Dyson equations is discussed, and the Hamiltonian structure for the homogeneous expansion of a free nonr otating ellipsoid is derived. (C) 2000 American Institute of Physics. [S002 2- 2488(00)01502-4].