Variational sequences in mechanics

Abstract.

Let $Y$ be a fibered manifold over a base manifold $X$. A differential 1-form $\rho$, defined on the $r$-jet prolongation of $J^r Y$, is said to be contact, if it vanishes along the $r$-jet prolongation $J^rY$ of every section $\gamma$ of $Y$. The notion of contactness is naturally extended to $k$-forms with $k\ge 1$. The contact forms define a subsequence of the De Rham sequence on $J^rY$. The corresponding quotient sequence is known as the rth order variational sequence. In this paper, the case of 1-dimensional base $X$ is considered. A simple proof is given of the fact that the rth order variational sequence is an acyclic resolution of the constant sheaf. Then the 1st order variational sequence is studied in detail. The quotient sheaves, as well as the quotient mappings, are determined explicitly, and their relationship to the standard concepts of the 1st order calculus of variations is discussed. The following is shown: a) the lagrangians in the 1st order variational sequence (classes of 1-forms) coincide with 2nd order lagrangians, affine in the second derivative variables, b) the concept of the Euler-Lagrange form is extended to 2-forms which are not necessarily variational, c) the concept of the Helmholtz-Sonin form is introduced as the class of an arbitrary 3-form, d) the well-known fundamental notions such as the Euler-Lagrange, and Helmholtz-Sonin mappings are represented by two arrows at the beginning of the variational sequence; this relates the global structure of the Euler-Lagrange mapping to the cohomology of $Y$, e) all the remaining classes of $k$-forms with $k\ge 3$, as well as the quotient mappings, are determined explicitly, f) a locally variational form is defined as a generalization of a symplectic form; locally variational forms, associated to a fixed Euler-Lagrange form, are characterized, and g) distributions associated with a locally variational form are described, and their relation to the Euler-Lagrange equations is studied. These results illustrate differences between finite order variational sequences and variational bicomplexes, based on infinite jet constructions.