CALCULUS OF FORMS ALONG A MAP ADAPTED TO
THE STUDY OF SECOND-ORDER DIFFERENTIAL EQUATIONS
W. Sarlet, E. Mart\'{\i}nez and A. Vandecasteele
Abstract. Earlier work, in which we discussed a special class of
differential forms associated to a given second-order vector field $\G$,
has led the roots for a much more extensive and general study of scalar and
vector-valued forms along the tangent bundle projection $\t:TM\rightarrow
M$ which we briefly review first. The classification of derivations of such
forms requires an additional ingredient, namely a non-linear connection.
Particularly important in the case of the connection coming from a
second-order vector field $\G$ are the appearance of a degree zero
derivation, called the dynamical covariant derivative, and of a type (1,1)
tensor field along $\t$, called the Jacobi endomorphism. We further discuss
the extension of this theory to time-dependent equations, for which the
general set-up is the calculus of forms along the map $\pi:\R\times TM
\rightarrow \R\times M$. All essential results of the autonomous case have
an analogue in the time-dependent situation, but there are a number of
technical complications which make it sometimes hard to decide about the
best possible approach. A number of applications are presented. The most
successful of these so far concerns a complete characterization of
separable equations with the aid of algebraic conditions which are directly
verifiable in practice.
Keywords. Forms along a map, derivations, second-order equations,
separability.
MS classification. 58A10, 53C05, 70D05.