Variational and Geometric Structures of Discrete Dirac 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.

26 pages; published online in Foundations of Computational Mathematics (2011)

References in corpus (6)

• Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids
• Geometric Hamilton-Jacobi Theory for Nonholonomic Dynamical Systems
• Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization
• Discrete Nonholonomic Lagrangian Systems on Lie Groupoids
• Nonholonomic Hamilton-Jacobi equation and Integrability
• Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids

Cited by in corpus (12)

• Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
• Retraction maps: a seed of geometric integrators
• Tensor products of Dirac structures and interconnection in Lagrangian mechanical systems
• The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories
• Multisymplectic Hamiltonian Variational Integrators
• Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds
• Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators
• Variational integrators for interconnected Lagrange-Dirac systems
• Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups
• Geometric Methods for Adjoint Systems
• Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics
• Variational Principles for Hamiltonian Systems