Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

TL;DR

This paper introduces novel discrete tangent bundle structures for variational discretizations of Lagrangian systems, enabling high-order, structure-preserving numerical integrators for complex mechanical systems.

Contribution

It develops alternative discrete tangent bundles extending tangent vectors to finite curve segments, facilitating advanced variational discretizations for nonholonomic and constrained systems.

Findings

Discrete tangent bundles enable high-order integrators.

Framework preserves geometric structures and symmetries.

Applicable to nonholonomic mechanical systems with nonlinear constraints.

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.