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Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

Marsden, Jerrold E. and Patrick, George W. and Shkoller, Steve (1998) Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs. Communications in Mathematical Physics, 199 (2). pp. 351-395. ISSN 0010-3616. doi:10.1007/s002200050505.

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This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes.

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Additional Information:© 1998 Springer-Verlag. Received February 1997; this version, January 3, 1998. We would like to extend our gratitude to Darryl Holm, Tudor Ratiu and Jeff Wendlandt for their time, encouragement and invaluable input. Work of J. Marsden was supported by the California Institute of Technology and NSF grant DMS 96-33161. Work by G. Patrick was partially supported by NSERC grant OGP0105716 and that of S. Shkoller was partially supported by the Cecil and Ida M. Green Foundation and DOE. We also thank the Control and Dynamical Systems Department at Caltech for providing a valuable setting for part of this work.
Funding AgencyGrant Number
NSFDMS 96-33161
Natural Sciences and Engineering Research Council (NSERC)OGP0105716
Cecil and Ida M. Green FoundationUNSPECIFIED
Department of Energy (DOE)UNSPECIFIED
Issue or Number:2
Record Number:CaltechAUTHORS:20100903-075958375
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19773
Deposited By: Ruth Sustaita
Deposited On:09 Sep 2010 16:54
Last Modified:08 Nov 2021 23:55

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