The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler--Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.Comment: 19 pages, minor changes, final version to appear in Diff. Geom. App

Geometric aspects of higher order variational principles on submanifolds

The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples are provided.Comment: 17 page