Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Abstract

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vectors. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.

Typos corrected. New abstract. Diagrams added. Some additional information and a conclusions section added

References in corpus (4)

• Discrete differential geometry. Consistency as integrability
• Skew critical problems
• Lagrangian mechanics without ordinary differential equations
• On converting any one-step method to a variational integrator of the same order

Cited by in corpus (5)

• Retraction maps: a seed of geometric integrators
• Error analysis of forced discrete mechanical systems
• On converting any one-step method to a variational integrator of the same order
• General Techniques for Constructing Variational Integrators
• Error analysis of variational integrators of unconstrained Lagrangian systems