On the Variational Characterisation of Generalized Jacobi Equations

TL;DR

This paper explores the variational structure of second-order Lagrangians, introduces new related Lagrangians, and shows how Jacobi equations can be derived from a unified variational principle, revealing conserved quantities.

Contribution

It introduces a hierarchy of Lagrangians for higher-order variational derivatives and demonstrates a unified variational approach to derive both Euler-Lagrange and Jacobi equations.

Findings

Jacobi equations can be obtained as variational equations from a new Lagrangian.

A hierarchy of Lagrangians relates different order variational derivatives.

Conservation of an energy-momentum tensor occurs when the original Lagrangian is space-time independent.

Abstract

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.