  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.
