Geodesic
Second variational derivative of local variational problems and conservation laws
Archivum mathematicum, Tome 47 (2011) no. 5, pp. 395-403 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We consider cohomology defined by a system of local Lagrangian and investigate under which conditions the variational Lie derivative of associated local currents is a system of conserved currents. The answer to such a question involves Jacobi equations for the local system. Furthermore, we recall that it was shown by Krupka et al. that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form; we remark that the corresponding local system of Euler–Lagrange forms is variationally equivalent to a global one.
We consider cohomology defined by a system of local Lagrangian and investigate under which conditions the variational Lie derivative of associated local currents is a system of conserved currents. The answer to such a question involves Jacobi equations for the local system. Furthermore, we recall that it was shown by Krupka et al. that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form; we remark that the corresponding local system of Euler–Lagrange forms is variationally equivalent to a global one.
Classification : 55N30, 55R10, 58A12, 58A20, 58E30, 70S10
Keywords: fibered manifold; jet space; Lagrangian formalism; variational sequence; second variational derivative; cohomology; symmetry; conservation law 
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Palese, Marcella; Winterroth, Ekkehart; Garrone, E. Second variational derivative of local variational problems and conservation laws. Archivum mathematicum, Tome 47 (2011) no. 5, pp. 395-403. http://geodesic.mathdoc.fr/item/ARM_2011_47_5_a6/

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