Mathematics – Differential Geometry
Scientific paper
1998-07-16
Mathematics
Differential Geometry
AMS-LaTeX, 19 pages, to appear in Math. Proc. Camb. Phil. Soc
Scientific paper
10.1017/S0305004198002953
This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. In this theory, solutions of a PDE are sections of a fiber bundle $Y$ over a base manifold $X$ of dimension $n$$+$1, typically taken to be spacetime. Given a connection on $Y$, a covariant Hamiltonian density ${\mathcal H}$ is then intrinsically defined on the primary constraint manifold $P_{\mathcal L}$, the image of the multisymplectic version of the Legendre transformation. One views $P_{\mathcal L}$ as a subbundle of $J^1(Y)^\star$, the affine dual of $J^1(Y)$, the first jet bundle of $Y$. A canonical multisymplectic ($n$$+$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$$+$1 2-forms $\omega^{(\mu)}$ defined by Bridges [1997] are a particular coordinate representation for a single multisymplectic ($n$$+$2)-form, and in the presence of symmetries, can be assembled into $\Omega_{\mathcal H}$. A generalized Hamiltonian Noether theory is then constructed which recovers the vanishing of the divergence of the vector of $n$$+$1 distinct momentum mappings defined in Bridges [1997] and, when applied to water waves, recovers Whitham's conservation of wave action. We also show the utility of this theory in the study of periodic pattern formation and wave instability.
Marsden Jerrold E.
Shkoller Steve
No associations
LandOfFree
Multisymplectic geometry, covariant Hamiltonians, and water waves does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Multisymplectic geometry, covariant Hamiltonians, and water waves, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Multisymplectic geometry, covariant Hamiltonians, and water waves will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-76796