Cohomological aspects of Noether theorem for Lagrangians of classical 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 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 arrive at the hierarchy in the space $\V_{0.0}$ corresponding to the cohomologies of the algebra $\G$ and configuration space $M$.

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Nota bene: The originally archived full text was mistakenly prepared from the TeX source of a different article by the authors, entitled "Double complexes and cohomological hierarchy in a space of weakly invariant Lagrangians of mechanics", of which an extended version appeared in Acta Appl. Math. 56, No. 2, 181-215 (1999). Neither the wrong nor the correct version appeared in the printed version which was published in 1998 as Volume 219 of the AMS series "Contemporary Mathematics" (ISBN 0-8218-0828-1).