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.