Variational Discretizations: Discrete Tangent Bundles, Local Error Analysis, and Arbitrary Order Variational Integrators

Discretizations of variational principles of physical systems are towards discrete models that have a theoretical status equivalent to the continuous models. Practically, such variational discretizations lead to a class of geometric numerical integrators, called variational integrators. To construct variational discretizations of Hamilton’s principle of mechanics, we develop geometric discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. In the formalism of constrained mechanics, such as that which underlies the SHAKE and RATTLE methods of molecular dynamics, we develop a method to convert any one‐step integrator to a variational integrator of the same order. Existence and uniqueness, and accuracy, of variational integrators, require due consideration of singularities at zero time‐step. We show existence and uniqueness for variational integrators by blowing up the variational principle at zero time‐step. The straight‐forward computation gives an accuracy one less than is observed in simulations, a deficit that is recovered by a past‐future symmetry of the blown‐up principle.