ON THE VARIATIONAL CHARACTERIZATION
OF GENERALIZED JACOBI EQUATIONS

Biagio Casciaro, Mauro Francaviglia and Victor Tapia

Key words: Calculus of variations, Jacobi equations.

1991 Math. Subject Classification: 70H20, 70H30.

Abstract: We study higher--order variational derivatives of a generic
second--order Lagrangian $\cL_0=\cL_0(x,\f,\p\f,\p^2\f)$ 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, $\cL_1$ and $\cL_2$, associated to the first and second--order
deformations of the original Lagrangian $\cL_0$ respectively; they are in
fact the first elements of a whole hierarchy of Lagrangians derived from
$\cL_0$. 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 $\cL_0$ 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 $\cL_1$ of the Lagrangian. We can furthermore
introduce an associated energy--momentum tensor ${\cH^\mu}_\nu$, which turns
out to be a conserved quantity if $\cL_0$ is independent of space--time
variables.