Abstract. In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are in a certain sense optimal, converging at the same rate as the best possible approximation in a certain function space. We further prove that certain geometric invariants also converge at an optimal rate, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence.
We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic, momentum preserving, and can be constructed to be of arbitrarily high-order, or can be made to converge geometrically. Furthermore, these methods are stable and accurate for very large time steps. We demonstrate the construction of one such variational integrator for the rigid body, and discuss how this construction could be generalized to other related Lie group problems. We close with several numerical examples which demonstrate our claims, and discuss further extensions of our work., which lead to the extension of other classical geometric structures. It is important to note that, while there are two different discrete Legendre transforms, (1) guarantees that FL − d (q k , q k+1 ) = FL + d (q k−1 , q k ), and thus they can be used interchangeably when defining the discrete geometric structure. By their construction, variational integrators induce a discrete symplectic form, i.e. Ω L d = (F ± L d )* Ω which is conserved by the update map F * L d Ω L d = Ω L d , and a discrete analogue of Noether's Theorem, which states that if a discrete Lagrangian is invariant under a diagonal group action on (q k , q k+1 ), it induces a discrete momentum map J L d = FL ± d * J, which is preserved under the update map: F * L d J L d = J L d . The existence of these discrete
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