Title: Discretizations of Lagrangian Mechanics

Abstract: We develop variational discretizations of mechanics, to the
generality of discretizing nonholonomic mechanical systems with
nonlinear constraints. This development is based on discrete analogues
of tangent bundles obtained by systematically extending tangent
vectors to finite curve segments. We show existence and uniqueness of
the discrete evolutions by blowing up the variational principles at
zero time-step.  The blown-up variational principles have a
past-future symmetry, which we use to prove the order of consistency
between the the continuous and discrete evolutions, a problem which
has been oversimplified in existing work. Our discretzation methods
can automatically convert any one-step numerical method to a
variational method of the same order.