  We study higher--order variational derivatives of a generic second--order
Lagrangian ${\cal L}={\cal L}(x,\phi,\partial\phi,\partial^2\phi)$ and in this
context we discuss the Jacobi equation ensuing from the second variation of the
action. We exhibit the different integrations by parts which may be performed
to obtain the Jacobi equation and we show that there is a particular
integration by parts which is invariant. We introduce two new Lagrangians,
${\cal L}_1$ and ${\cal L}_2$, associated to the first and second--order
deformations of the original Lagrangian ${\cal L}_0$ respectively; they are in
fact the first elements of a whole hierarchy of Lagrangians derived from ${\cal
L}_0$. In terms of these Lagrangians we are able to establish simple relations
between the variational derivatives of different orders of a given Lagrangian.
We then show that the Jacobi equations of ${\cal L}_0$ may be obtained as
variational equations, so that the Euler--Lagrange and the Jacobi equations are
obtained from a single variational principle based on the first--order
variation ${\cal L}_1$ of the Lagrangian. We can furthermore introduce an
associated energy--momentum tensor ${{\cal H}^\mu}_\nu$ which turns out to be a
conserved quantity if ${\cal L}_0$ is independent of space--time variables.
