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

Han-Ying Guo·Jianzhong Pan·Bin Zhou
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Abstract

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, ω)$, 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.

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Han-Ying Guo, Jianzhong Pan, Bin Zhou. 2004-08-22. The Generalized Liouville's Theorems via Euler-Lagrange Cohomology Groups on Symplectic Manifold. https://arxiv.org/abs/math-ph/0408034

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