Topology of the isoenergy manifolds of the Kirchhoff rigid body case on e(3)

Abstract

Our aim is to study topology of the Kirchhoff case of rigid body motion in an ideal incompressible fluid. We introduce the reduced potential for general Hamiltonian systems on e(3) with mixed quadratic terms. In application to the Kirchhoff case, we describe the Reeb graphs of the reduced potential. We provide a complete topological description of the three-dimensional isoenergy manifolds for that system, based on a combinatorial study of the Reeb graphs. Studying its momentum map, we describe the points of ranks zero and one. The Poincaré model of a rigid body with an ellipsoidal cavity filled with an ideal incompressible liquid has a Hamiltonian of the same form as the Kirchhoff Hamiltonian, with the underlying Poisson algebra being so(4). A similar analysis of bifurcations of the momentum map is presented in the Poincaré case as well.