arXiv subjects
Publications and source records attributed to Eugenio A. M. Rocha.
We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether's first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2,3,5,8).
The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally differentiable Lagrangians. Here, we formulate a Legendre condition and an extension of the classical theorem of Emmy Noether, thus obtaining first integrals for problems of the calculus of variations on locally convex spaces.
We obtain a method to compute effective first integrals by combining Noether's principle with the Kozlov-Kolesnikov integrability theorem. A sufficient condition for the integrability by quadratures of optimal control problems with controls taking values on open sets is obtained. We illustrate our approach on some problems taken from the literature. An alternative proof of the integrability of the sub-Riemannian nilpotent Lie group of type (2,3,5) is also given.