A Frame Bundle Generalization of Multisymplectic Momentum Mappings

TL;DR

This paper develops a generalized framework for momentum mappings in covariant Hamiltonian field theories, extending symplectic geometry to a bundle of linear frames, leading to new conserved quantities and transformation laws.

Contribution

It introduces a novel generalization of symplectic geometry to the bundle of vertically adapted frames, providing new momentum observables and conservation laws in field theories.

Findings

Generalized momentum mappings arise from automorphisms of the field configuration bundle.

The new framework produces conserved quantities for symmetries like translation and rotation.

It recovers classical results such as the parallel axis theorem and Lorentz transformations.

Abstract

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, 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 $L_VY$ 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 $L_VY$. Furthermore, momentum observables on $L_VY$ produce conserved quantities along flows in $L_VY$. 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.