Variational and Geometric Structures of Discrete Dirac Mechanics

TL;DR

This paper develops a geometric framework for discrete Dirac mechanics, enabling the derivation of integrators for constrained and degenerate systems, unifying Lagrangian and Hamiltonian approaches.

Contribution

It introduces discrete analogues of Tulczyjew's triple and Dirac structures, providing a unified geometric foundation for discrete Lagrangian and Hamiltonian mechanics with constraints.

Findings

Constructed discrete Tulczyjew's triple and Dirac structures.

Derived discrete Lagrange-Dirac and nonholonomic Hamiltonian integrators.

Unified treatment of discrete Lagrangian and Hamiltonian mechanics.

Abstract

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.