An alternative form of the Helmholtz criterion in the inverse problem of calculus of variations
arXiv:math/0308179 · doi:10.1023/B:MATH.0000032834.91846.02
Abstract
We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its relation with the classical Helmholtz criterion using the de Rham complex on an infinite-dimensional space of fields.
8 pages. LaTeX 2e. New revised version (revision of exposition in some places, introduction added, style edited, one statement added)