Cubic polynomials on Lie groups: reduction of the Hamiltonian system

Abstract

This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden–Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie–Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.