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 on critical sections is equal to the vertical differential of the Euler--Lagrange morphism and (up to total divergencies) to its adjoint morphism. These two objects generalize in an invariant way the Hessian and the Jacobi morphisms of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.
A geometric formulation of Hessian and Jacobi tensors for higher order Lagrangians
VITOLO, Raffaele
2005-01-01
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 on critical sections is equal to the vertical differential of the Euler--Lagrange morphism and (up to total divergencies) to its adjoint morphism. These two objects generalize in an invariant way the Hessian and the Jacobi morphisms of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
