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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., Comment: This article was originally published in Differential Geometry and Applications, Proc. Conf., Aug. 28 - Sept. 1, 1995, Brno, Czech Republic (Masaryk University, Brno, 1996), pp. 353-372. Since it is hardly available we post it here for the interested audience

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., Comment: This article was originally published in Differential Geometry and Applications, Proc. Conf., Aug. 28 - Sept. 1, 1995, Brno, Czech Republic (Masaryk University, Brno, 1996), pp. 353-372. Since it is hardly available we post it here for the interested audience

We consider the second variation and the appropriate Jacobi eqautions for the scalar curvature and the quadratic curvature invariants based on an independent pair (g, T) formed by a metric and torsionless linear connection (so-called 'Palatini formalism'). The purely metric case is obtained as a consequence. The results are worked out in fill detail in view of applications to non-linear Lagrangian theories of gravitation.

We consider the second variation and the appropriate Jacobi eqautions for the scalar curvature and the quadratic curvature invariants based on an independent pair (g, T) formed by a metric and torsionless linear connection (so-called 'Palatini formalism'). The purely metric case is obtained as a consequence. The results are worked out in fill detail in view of applications to non-linear Lagrangian theories of gravitation.