  Let $M$ be smooth $n$-dimensional manifold, fibered over a $k$-dimensional
submanifold $B$ as $\pi:M \to B$, and $\vartheta \in \Lambda^k (M)$; one can
consider the functional on sections $\phi$ of the bundle $\pi$ defined by
$\int_D \phi^* (\vartheta)$, with $D$ a domain in $B$. We show that for $k =
n-2$ the variational principle based on this functional identifies a unique (up
to multiplication by a smooth function) nontrivial vector field in $M$, i.e. a
system of ODEs. Conversely, any vector field $X$ on $M$ satisfying $i_X ({\rm
d} \vartheta) = 0$ for some $\vartheta \in \Lambda^{n-2} (M)$ admits such a
variational characterization. We consider the general case, and also the
particular case $M = P \times R$ where one of the variables (the time) has a
distinguished role; in this case our results imply that any Liouville
(volume-preserving) vector field on the phase space $P$ admits a variational
principle of the kind considered here.
