On the Variational Characterization of Generalized Jacobi Equations

We study higher{ordervariational derivativesof a generic second{order Lagrangian L 0 = L 0 (x; ; @; @ 2) and in this context we discuss the Jacobi equation ensuing from the second variation of the action. We exhibit the diierent inte-grations 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, L 1 and L 2 , associated to the rst and second{order deformations of the original Lagrangian L 0 respectively; they are in fact the rst elements of a whole hierarchy of Lagrangians derived from 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 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 rst{ order variation L 1 of the Lagrangian. We can furthermore introduce an associated energy{momentum tensor H , which turns out to be a conserved quantity if L 0 is independent of space{time variables. 0. Introduction As is well known, the second variation of an action functional governs the behaviour of the action itself in the neighbourhood of critical sections. In particular, the Hessian of the Lagrangian deenes a quadratic form whose sign properties allow to distinguish between minima, maxima and degenerate critical sections (see, e.g., 1]). It is also well known that in the case of geodesics in a Riemannian manifold those elds which govern the transition from geodesics to geodesics, (i.e. those vectorrelds which make the second variation to vanish identically modulo boundary terms) are called Jacobi elds 2] and they are solutions of a second{order diierential equation known as Jacobi equation (of geodesics). The notion of Jacobi equation as an outcome of the second variation is in fact fairly more general than this and general formulae for the second variation and