The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations

TL;DR

This paper develops a geometric framework for higher order calculus of variations, generalizing Hessian and Jacobi morphisms, and shows their self-adjointness properties along solutions.

Contribution

It introduces an invariant geometric formalism for higher order variations, extending classical Hessian and Jacobi morphisms to complex Lagrangians with multiple derivatives.

Findings

Second variation equals the vertical differential of the Euler-Lagrange morphism.

The Euler-Lagrange morphism is self-adjoint along solutions.

Provides examples illustrating the geometric formalism.

Abstract

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.