Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids

TL;DR

This paper introduces a discrete Hamilton-Pontryagin variational principle on Lie groupoids, unifying two approaches to discrete Lagrangian mechanics and providing a framework for geometric integration.

Contribution

It develops a discrete variational principle on Lie groupoids that unifies existing methods and includes explicit examples like the pair groupoid.

Findings

Unified framework for discrete Lagrangian mechanics on Lie groupoids

Demonstrated equivalence of variational and generating function approaches

Provided explicit example with the pair groupoid

Abstract

We present a discrete analog of the recently introduced Hamilton-Pontryagin variational principle in Lagrangian mechanics. This unifies two, previously disparate approaches to discrete Lagrangian mechanics: either using the discrete Lagrangian to define a finite version of Hamilton's action principle, or treating it as a symplectic generating function. This is demonstrated for a discrete Lagrangian defined on an arbitrary Lie groupoid; the often encountered special case of the pair groupoid (or Cartesian square) is also given as a worked example.