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Francaviglia, Mauro; Palese, Marcella; Vitolo, Raffaele

The Hessian and Jacobi morphisms for higher order calculus of variations. (English) Zbl 1065.58010

Differ. Geom. Appl. 22, No. 1, 105-120 (2005).
Summary: 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.

Cited in 9 Documents

MSC:

58E30 Variational principles in infinite-dimensional spaces
58A20 Jets in global analysis

Keywords:

Fibered manifold; Jet space; Variational sequence; Second variation
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References:

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