Generalizing Hamiltonian Mechanics with Closed Differential Forms

TL;DR

This paper extends classical Hamiltonian mechanics to systems with multiple invariants using multisymplectic geometry, establishing a new geometric framework that generalizes the traditional theory and connects it to Nambu mechanics.

Contribution

It introduces a novel correspondence between generalized Hamiltonian systems with multiple invariants and multisymplectic geometry, expanding the geometric understanding of conserved quantities.

Findings

Classical Hamiltonian systems with multiple invariants are also generalized Hamiltonian systems.

A generalized Hamiltonian system with multiple invariants corresponds to a classical system on a level set of invariants.

Reinterpretation of Lie-Darboux and Liouville theorems within the multisymplectic framework.

Abstract

Classical Hamiltonian mechanics, characterized by a single conserved Hamiltonian (energy) and symplectic geometry, `hides' other invariants into symmetries of the Hamiltonian or into the kernel of the Poisson tensor. Nambu mechanics aims to generalize classical Hamiltonian mechanics to ideal dynamical systems bearing two Hamiltonians, but its connection to a suitable geometric framework has remained elusive. This work establishes a novel correspondence between generalized Hamiltonian mechanics, defined for systems with a phase space conservation law (invariance of a closed form) and a matter conservation law (invariance of multiple Hamiltonians), and multisymplectic geometry. The key lies in the invertibility of differential forms of degree higher than 2. We demonstrate that the cornerstone theorems of classical Hamiltonian mechanics (Lie-Darboux and Liouville) require reinterpretation within this new framework, reflecting the unique properties of invertibility in multisymplectic geometry. Furthermore, we present two key theorems that solidify the connection: i) any classical Hamiltonian system with two or more invariants is also a generalized Hamiltonian system and ii) given a generalized Hamiltonian system with two or more invariants, there exists a corresponding classical Hamiltonian system on the level set of all but one invariant, with the remaining invariant playing the role of the Hamiltonian function.