Variational and Geometric Structures of Discrete Dirac Mechanics

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 symple...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 11; no. 5; pp. 529 - 562
Main Authors: Leok, Melvin, Ohsawa, Tomoki
Format: Journal Article
Language:English
Published: New York Springer-Verlag 01.10.2011
Springer Nature B.V
Subjects:
Applications of Mathematics
Categories
Computational mathematics
Computer Science
Derivation
Economics
Foundations
Geometry
Integrators
Lagrange multiplier
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Variational principles
65P10
Lagrange–Dirac systems
37J60
Dirac structures
70H45
70F25
Geometric integration
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-011-9096-2