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 [Hscr ] is then
intrinsically defined on
the primary constraint manifold P[Lscr ], the image
of the multisymplectic version of the
Legendre transformation. One views P[Lscr ] as a subbundle
of J1(Y)*, the affine dual of
J1(Y)*, the first jet bundle of Y.
A canonical multisymplectic (n+2)-form Ω[Hscr ]
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[Hscr ]. A generalized Hamiltonian
Noether theory is
then constructed which relates the action of the symmetry groups lifted
to P[Lscr ] 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.