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Variational problems with fractional derivatives: Invariance conditions and Nöther’s theorem

✍ Scribed by Teodor M. Atanacković; Sanja Konjik; Stevan Pilipović; Srboljub Simić

Publisher: Elsevier Science
Year: 2009
Tongue: English
Weight: 813 KB
Volume: 71
Category: Article
ISSN: 0362-546X
DOI: 10.1016/j.na.2008.12.043

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✦ Synopsis

A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal transformation group (basic Nöther's identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of Nöther's theorem leading to conservation laws for fractional Euler-Lagrangian equation is obtained as well. Results are illustrated by several concrete examples. Finally, an approximation of a fractional Euler-Lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws.

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Necessary optimality conditions for frac

Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β)

✍ Rami Ahmad El-Nabulsi; Delfim F. M. Torres 📂 Article 📅 2007 🏛 John Wiley and Sons 🌐 English ⚖ 110 KB

## Abstract We derive Euler–Lagrange‐type equations for fractional action‐like integrals of the calculus of variations which depend on the Riemann–Liouville derivatives of order (α, β), α>0, β>0, recently introduced by Cresson. Some interesting consequences are obtained and discussed. Copyright © 2