A REPRESENTATION OF THE VARIATIONAL SEQUENCE
IN HIGHER ORDER MECHANICS

JAROSLAV \v STEF\'ANEK

Key words: Fibered manifold, jet, contact form, variational sequence.

1991 Math. Subject Classification: 49F05, 58A15, 58A20, 58E99.

Abstract: Let $Y$ be a fibered manifold over a base manifold $X$.
A differential form $\rho$ defined on the $r$-jet prolongation
$J^rY$ of $Y$ is said to be {\it contact} if it vanishes along
the $r$-jet prolongation $J^r\gamma$ of every section $\gamma$
of $Y$. The contact forms define a subsequence of the de Rham
sequence on $J^rY$. The corresponding quotient sequence is known as the
$r$-{\it th order variational sequence}. We consider the case of
$1$-dimensional base $X$ ({\it mechanics}).

We construct an isomorphism between the variational sequence and another
sequence whose elements are spaces of forms rather than spaces
of classes of forms. In this way we determine explicitly the first
two mappings of the variational sequence.