On the geometry of Riemannian cubic polynomials

AbstractWe continue the work of Crouch and Silva Leite on the geometry of cubic polynomials on Riemannian manifolds. In particular, we generalize the theory of Jacobi fields and conjugate points and present necessary and sufficient optimality condition

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

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.Comment: 20 pages. Final version which incorporates the Corrigendum recently published (J. Phys. A: Math. Theor. 46 189501, 2013

A second order Riemannian variational problem from a Hamiltonian perspective

We present a Hamiltonian formulation of a second order variational problem on a differentiable manifold Q, endowed with a Riemannian metric and explore the possibility of writing down the extremal solutions of that problem as a flow in the space TQ T*Q T*Q. For that we utilize the connection r on Q, corresponding to the metric . In general the results depend upon a choice of frame for TQ, but for the special situation when Q is a Lie group G with Lie algebra G, our results are global and the flow reduces to a flow on G x G x G* x G*