The Generalized Liouville's Theorems via Euler-Lagrange Cohomology Groups on Symplectic Manifold

TL;DR

This paper generalizes Liouville's theorem and Noether's theorem on symplectic manifolds using Euler-Lagrange cohomology groups, providing a broader classification of vector fields and conservation laws in classical mechanics.

Contribution

It introduces a new framework using Euler-Lagrange cohomology groups to generalize classical theorems on symplectic manifolds, linking cohomology directly to conservation laws.

Findings

Generalized Liouville's theorems to symplectic-like and Hamiltonian-like cases

Established a classification of vector fields via Euler-Lagrange cohomology

Connected cohomology sequences to conservation laws and symmetries

Abstract

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, \omega)$, their properties and a kind of classification of vector fields on the manifold, we generalize Liouville's theorem in classical mechanics to two sequences, the symplectic(-like) and the Hamiltonian-(like) Liouville's theorems. This also generalizes Noether's theorem, since the sequence of symplectic(-like) Liouville's theorems link to the cohomology directly.