[Home]
[
News]
[ Events]
[ People]
[ Research]
[ Education]
[Visitor Info]
[UCSD Only]
[Admin]
Discrete Dirac Structures and Variational Discrete Dirac Mechanics
by Melvin Leok, Tomoki Ohsawa
Abstract:
We construct discrete analogues of Dirac structures by considering the geometry of symplectic maps and their associated generating functions, in a manner analogous to the construction of continuous Dirac structures in terms of the geometry of symplectic vector fields and their associated Hamiltonians. We demonstrate that this framework provides a means of deriving implicit discrete Lagrangian and Hamiltonian systems, while incorporating discrete Dirac constraints. In particular, this yields implicit nonholonomic Lagrangian and Hamiltonian integrators. We also introduce a discrete Hamilton-Pontryagin variational principle on the discrete Pontryagin bundle, which provides an alternative derivation of the same set of integration algorithms. In so doing, we explicitly characterize the discrete Dirac structures that are preserved by Hamilton-Pontryagin integrators. In addition to providing a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of Dirac mechanics, it provides a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators. Since discrete Lagrangians and discrete Hamiltonians are essentially generating functions of different types, the theoretical framework described in this paper is sufficiently general to encompass all possible Dirac integrators through an appropriate choice of generating functions.
UCSD-CCoM-08-07.pdf October 2008
News]
[ Events]
[ People]
[ Research]
[ Education]
[Visitor Info]
[UCSD Only]
[Admin]
Directors:
Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
[ Click to Send Email ]
Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
[ Click to Send Email ]
by Melvin Leok, Tomoki Ohsawa
Abstract:
We construct discrete analogues of Dirac structures by considering the geometry of symplectic maps and their associated generating functions, in a manner analogous to the construction of continuous Dirac structures in terms of the geometry of symplectic vector fields and their associated Hamiltonians. We demonstrate that this framework provides a means of deriving implicit discrete Lagrangian and Hamiltonian systems, while incorporating discrete Dirac constraints. In particular, this yields implicit nonholonomic Lagrangian and Hamiltonian integrators. We also introduce a discrete Hamilton-Pontryagin variational principle on the discrete Pontryagin bundle, which provides an alternative derivation of the same set of integration algorithms. In so doing, we explicitly characterize the discrete Dirac structures that are preserved by Hamilton-Pontryagin integrators. In addition to providing a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of Dirac mechanics, it provides a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators. Since discrete Lagrangians and discrete Hamiltonians are essentially generating functions of different types, the theoretical framework described in this paper is sufficiently general to encompass all possible Dirac integrators through an appropriate choice of generating functions.
UCSD-CCoM-08-07.pdf October 2008