Nonshifted calculus of variations on time scales with ∇-differentiable σReport as inadecuate

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Abstract : In calculus of variations on general time scales, an Euler-Lagrange equation of integral form is usually derived in order to characterize the critical points of nonshifted Lagrangian functionals, see e.g. R.A.C. Ferreira and co-authors, Optimality conditions for the calculus of variations with higher-order delta derivatives, Appl. Math. Lett., 2011. In this paper, we prove that the ∇-differentiability of the forward jump operator σ is a sharp assumption on the time scale in order to ∇-differentiate this integral Euler-Lagrange equation. This procedure leads to an Euler-Lagrange equation of differential form. Furthermore, from this differential form, we prove a Noether-type theorem providing an explicit constant of motion for Euler-Lagrange equations admitting a symmetry.

Keywords : Euler-Lagrange equations Noether-s theorem Time scale calculus of variations

Author: Loïc Bourdin -



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