Double Complexes and Cohomological Hierarchy in a Space of Weakly Invariant Lagrangians of Mechanics

Abstract

For a given configuration space $M$ and Lie algebra $g$ whose action is defined on $M$, the space $V_{0.0}$ of weakly $g$-invariant Lagrangians (i.e. Lagrangians whose motion equations left hand sides are $g$-invariant) is studied. The problem is reformulated in the terms of the double complex of Lie algebra cochains with values in the complex of Lagrangians. Calculating the cohomology of this complex using the method of spectral sequences, we come to the hierarchy in the space $V_{0.0}$: The double filtration $V_{s.r}$ ($s=0,1,2,3,4;r=0,1$) and the homomorphisms on every space $V_{s.r}$ are constructed. These homomorphisms take values in cohomologies of the Lie algebra $g$ and configuration space $M$. On one hand every space $V_{s.r}$ is the kernel of the corresponding homomorphism, on the other hand this space is defined by its physical properties.