A Frame Bundle Generalization of Multisymplectic Momentum Mappings

Abstract

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_{V}Y$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum observables are vector-valued momentum mappings on the vertically adapted frame bundle generated from automorphisms of $Y$. The generalized symplectic geometry on $L_{V}Y$ is a covering theory for multisymplectic geometry on the multiphase space $Z$, and it follows that the field momentum observables on $Z$ are generalized by those on $L_{V}Y$. Furthermore, momentum observables on $L_{V}Y$ produce conserved quantities along flows in $L_{V}Y$. For translational and orthogonal symmetries of fields and reparametrization symmetry in mechanics, momentum is conserved, and for angular momentum in time-evolution mechanics we produce a version of the parallel axis theorem of rotational dynamics, and in special relativity, we produce the transformation of angular momentum under boosts.

23 pages

References in corpus (2)

• Multisymplectic geometry, variational integrators, and nonlinear PDEs
• A frame bundle generalization of multisymplectic field theories