A second order Riemannian variational problem from a Hamiltonian perspective

• Author(s): Crouch, P. ; Leite, F. Silva ; Camarinha, M.
• Date: 1998
• Persistent ID: http://hdl.handle.net/10316/11230
• Origin: Estudo Geral - Universidade de Coimbra
• Subject(s): Riemannian manifolds; Lie groups; Hamiltonian equations; Optimal control; Variational problems
• Document Type: Preprint
• Language: English

Description

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*.