Physics – Mathematical Physics
Scientific paper
2003-11-13
Diff. Geom. Appl. 22(1) (2005) 105-120.
Physics
Mathematical Physics
19 pages, minor changes, final version to appear in Diff. Geom. Appl
Scientific paper
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.
Francaviglia Mauro
Palese Marcella
Vitolo Raffaele
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