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Multisymplectic geometry, covariant Hamiltonians, and water waves

Marsden, Jerrold E. and Shkoller, Steve (1999) Multisymplectic geometry, covariant Hamiltonians, and water waves. Mathematical Proceedings of the Cambridge Philosophical Society, 125 (3). pp. 553-575. ISSN 0305-0041. doi:10.1017/S0305004198002953.

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This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by Bridges. In this theory, solutions of a partial differential equation are sections of a fibre bundle Y over a base manifold X of dimension n+1, typically taken to be spacetime. Given a connection on Y, a covariant Hamiltonian density [script H] is then intrinsically defined on the primary constraint manifold P_[script L], the image of the multisymplectic version of the Legendre transformation. One views P_[script L] as a subbundle of J^1(Y)^*, the affine dual of J^1(Y)^*, the first jet bundle of Y. A canonical multisymplectic (n+2)-form Ω_[script H] is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original partial differential equation as well as the Euler–Lagrange equations of the corresponding Lagrangian. Furthermore, we show that the n+1 2-forms ω^(µ) defined by Bridges are a particular coordinate representation for a single multisymplectic (n+2)-form and, in the presence of symmetries, can be assembled into Ω_[script H]. A generalized Hamiltonian Noether theory is then constructed which relates the action of the symmetry groups lifted to P_[script L] with the conservation laws of the system. These conservation laws are defined by our generalized Noether's theorem which recovers the vanishing of the divergence of the vector of n+1 distinct momentum mappings defined by Bridges and, when applied to water waves, recovers Whitham's conservation of wave action. In our view, the multisymplectic structure provides the natural setting for studying dispersive wave propagation problems, particularly the instability of water waves, as discovered by Bridges. After developing the theory, we show its utility in the study of periodic pattern formation and wave instability.

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Additional Information:© Cambridge Philosophical Society 1999. Received 24 March 1997; revised 1 September 1997. The authors would like to thank Tom Bridges and Mark Gotay for reading early drafts of the manuscript and making numerous comments and suggestions. S. S. was partly supported by a fellowship from the Cecil H. and Ida M. Green Foundation. J. E. M. was partly supported by the National Science Foundation under Grant DMS-DMS-96 33161 and the Department of Energy under Contract DE-FG0395-ER25251.
Funding AgencyGrant Number
Cecil H. and Ida M. Green FoundationUNSPECIFIED
NSFDMS-DMS-96 33161
Department of EnergyDE-FG0395-ER25251
Issue or Number:3
Record Number:CaltechAUTHORS:20100914-084439105
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19927
Deposited By: Ruth Sustaita
Deposited On:15 Sep 2010 20:40
Last Modified:08 Nov 2021 23:56

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