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Generalizing Hamiltonian Mechanics with Closed Differential Forms

Nathan Duignan, Naoki Sato

TL;DR

The paper develops a generalized Hamiltonian framework for degree $k>2$, where phase-space conservation is encoded by a closed $k$-form $w$ and matter invariants by $k-1$ Hamiltonians via the HDW equation $\iota_X w = -dH^1\wedge\dots\wedge dH^{k-1}$, with $dw=0$. It establishes a precise link to multisymplectic geometry through the invertibility of higher-degree forms, and proves two key correspondences: (i) any classical Hamiltonian system with two or more invariants is a generalized Hamiltonian system; (ii) a generalized system with two or more invariants yields a classical Hamiltonian system on level sets of all but one invariant. The work further develops Lie-Darboux-type structure, Liouville measures on reduced manifolds, and clarifies the role of the Jacobi identity and invariant measures in this generalized setting, including when FI fails in Nambu-type formalisms. Concrete examples—quasisymmetric magnetic fields, semiclassical quantum oscillators, and a reduction theorem—illustrate the framework and its potential to unify and extend classical and Nambu-type dynamics with applications in physics.

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.

Generalizing Hamiltonian Mechanics with Closed Differential Forms

TL;DR

The paper develops a generalized Hamiltonian framework for degree , where phase-space conservation is encoded by a closed -form and matter invariants by Hamiltonians via the HDW equation , with . It establishes a precise link to multisymplectic geometry through the invertibility of higher-degree forms, and proves two key correspondences: (i) any classical Hamiltonian system with two or more invariants is a generalized Hamiltonian system; (ii) a generalized system with two or more invariants yields a classical Hamiltonian system on level sets of all but one invariant. The work further develops Lie-Darboux-type structure, Liouville measures on reduced manifolds, and clarifies the role of the Jacobi identity and invariant measures in this generalized setting, including when FI fails in Nambu-type formalisms. Concrete examples—quasisymmetric magnetic fields, semiclassical quantum oscillators, and a reduction theorem—illustrate the framework and its potential to unify and extend classical and Nambu-type dynamics with applications in physics.

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.

Paper Structure

This paper contains 13 sections, 12 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Conventions on pairing and contraction of forms and multi-vectors
  3. Generalized Hamiltonian Mechanics
  4. Definition of some generalized Hamiltonian systems
  5. Invertibility and Poisson tensors
  6. Correspondence between classical Hamiltonian mechanics and generalized Hamiltonian mechanics of degree $k$
  7. Form flatness, Lie-Darboux theorems, and Liouville measures
  8. Remarks on the Jacobi identity and invariant measures in generalized Hamiltonian mechanics
  9. Examples
  10. Quasisymmetry as a generalized Hamiltonian system
  11. Semiclassical quantum oscillators
  12. An example of application of Theorem \ref{['thm:classicalFromGeneralised']}
  13. Concluding remarks

Key Result

Theorem 3.1

Let $\omega\in\Omega^2(M)$, consider the functions $H, C^1,\dots, C^{k-2}\in C^{\infty}(M)$, and define $dC := dC^1\wedge\dots\wedge dC^{k-2} \in \Omega^{k-2}(M)$. If there exists a vector field $X$ on $M$ such that then, the $k$-form satisfies That is, $X$ is a generalised Hamiltonian vector field of $w$ with Hamiltonians $H,C^1,\dots,C^{k-2}$.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 28 more