Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

This paper gives necessary and sufficient conditions for the (n-dimensional) generalized free rigid body to be in a state of relative equilibrium. The conditions generalize those for the case of the three-dimensional free rigid body, namely that the body is in relative equilibrium if and only if its angular velocity and angular momentum align, that is, if the body rotates about one of its principal axes. For the n-dimensional rigid body in the Manakov formulation, these conditions have a similar interpretation. We use this result to state and prove a generalized Saari's Conjecture (usually stated for the N -body problem) for the special case of the generalized rigid body.

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We present preliminary results for a prequantization procedure that leads in a natural way to the Dirac equation. The starting point is the recently introduced n-symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M in which the Rn-valued soldering 1-form 0 on LM plays the role of the n-symplectic potential. On a 4-dimensional spacetime manifold we consider the tensorial R4@ •4-valued function ~ on LM determined by the spacetime metric tensor g as the Hamiltonian for free observers and determine the associated 0~4-valued Hamiltonian vector field )?~ = Xi~| "Integration" of the X~ yields the dynamics of free observers on spacetime, namely parallel transport of linear frames along spacetime geodesics. In order to obtain a vector field on the spin bundle SM which is a lift of .~ and which is induced by a vector field .~ for an appropriate mapping ~, it is useful to define a prolongation L~ of some bundle L~ of oriented frames of M. If GL +(4, R) denotes the identity component of GL(4~, ~), then GL +(4, R) is the structure group of L~ and its double cover GL+(4, R) is the structure group of L'~. We show that the lift ~ of 0 on L~ to L~ induces a natural 4-sympleetic potential on L~ If ~ is the lift of ~ to L~ then we find the R4-valued Hamiltonian vector field .~ on L~ determined by ~ and show that the vector fields X~ on L'~ are tangent to the subbundle SM. "Integration" of the restriction of the X~ to SM now yields parallel transport of spin frames and thus tetrads along spacetime ~odesics of g. We consider a naive prequantization operator assignment X~,~,=ih?~X~ acting on C4-spinors in the standard representation of SL(2, C). The eigenvalue equation for the system of new Hilbert space operators yields the Dirac equation.

This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits a generalized symplectic structure from the full frame bundle. The geometric structure of the vertically adapted frame bundle admits vector-valued field observables and produces vector-valued Hamiltonian vector fields, from which we can define a Poisson bracket on the field observables. We show that the linear and affine multivelocity spaces and multiphase spaces for geometric field theories are associated to the vertically adapted frame bundle. In addition, the new geometry not only generalizes both the linear and the affine models of multisymplectic geometry but also resolves fundamental problems found in both multisymplectic models.

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