Euler-Lagrange Cohomology, Volume-Preserving Systems and Conservation Laws

摘要

Based on the Euler-Lagrange cohomology groups and the corresponding classification of the vector fields on a symplectic manifold (M,ω), we generalize famous Liou-ville's theorem in classical mechanics to a couple of symplectic(-like) and Hamiltonian-(like) sequences of Liouville's theorems. This also generalizes famous Noether's theorem, since the sequence of symplectic-like Liouville's theorems do not require the equations of motion globally.