Partial Differential Hamiltonian Systems

TL;DR

This paper introduces a novel formalism for Hamiltonian systems in field theory, unifying geometric structure and dynamics into a single framework, and explores their fundamental properties.

Contribution

It develops a new formalism for partial differential Hamiltonian systems that integrates geometric and dynamical aspects into one structure, differing from traditional multisymplectic approaches.

Findings

Defines PD Hamilton equations and proves their properties

Establishes a PD Noether theorem in the new formalism

Introduces a PD Poisson bracket compatible with the framework

Abstract

We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field theory, in our formalism, the geometric structure (kinematics) and the dynamical information on the "phase space" appear as just different components of one single geometric object.