Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian L : T (k) Q → R with k ≥ 1, the resulting discrete equations define a generally implicit numerical integrator algorithm on T (k−1) Q × T (k−1) Q that approximates the flow of the higher-order Euler-Lagrange equations for L. The algorithm equations are called higher-order discrete Euler-Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton's principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether's theorem, the momentum map.We construct an exact discrete Lagrangian L e d using the locally unique solution of the higher-order Euler-Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of L e d , we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.
Contents 24 References 24Mathematics Subject Classification (2010): 70G45, 70Hxx, 49J15.