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

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

2004-08-22

arxiv.org/abs/math-ph/0408034v1

Physics

Mathematical Physics

27 pages, no figure, revtex4

Scientific paper

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.

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