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

Abstract

For a given configuration space M and a Lie algebra G acting on M, the space V0.0 of weakly G-invariant Lagrangians, i.e., Lagrangians whose motion equations left-hand sides are G-invariant, is studied. The problem is reformulated in terms of the double complex of Lie algebra cochains with values in the complex of Lagrangians. Calculating the cohomology of this complex by the method of spectral sequences, we arrive at the hierarchy in the space V0.0: The double filtration {Vs.σ}, s = 0, 1, 2, 3, 4, σ = 0, 1, and the homomorphisms on every space {Vs.σ} are constructed. These homomorphisms take values in the cohomologies of the algebra G and the configuration space M. Every space {Vs.σ} is the kernel of the corresponding homomorphism, while the space itself is defined by its physical properties.