Abstract
In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques forsecond-order variational problems on Lie groupoids and their applications to the construction of variational integratorsfor optimal control problems of mechanical systems. Next, weshow how Lagrangian submanifolds of a symplectic groupoidgives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in a Lagrangian submanifold. We also study the theory ofreduction by symmetries and the corresponding Noether theorem.
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