Multisymplectic Geometry, Covariant Hamiltonians, and Water Waves

Marsden, J. E., S. Shkoller

Math. Proc. Camb. Phil. Soc. 125, 553-575.

Abstract:

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 1997.

In this theory, solutions of a PDE are sections of a fiber 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 $\mathcal {H}$ is then intrinsically defined on the primary constraint manifold P_{$\mathcal {L}$}, the image of the multisymplectic version of the Legendre transformation. One views P_{$\mathcal {L}$} as a subbundle of J¹(Y)^$\star$, the affine dual of J¹(Y), the first jet bundle of Y. A canonical multisymplectic (n+2)-form $\Omega_{\mathcal H}^{}$ is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangian.

We show that the n+1 2-forms $\omega^{(\mu)}_{}$ 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 $\Omega_{\mathcal H}^{}$. A generalized Hamiltonian Noether theory is then constructed which relates the action of the symmetry groups lifted to P_{$\mathcal {L}$} 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.

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.