A formulation of Noether's theorem for fractional problems of the calculus of variations

• Published in: Journal of mathematical analysis and applications Vol. 334; no. 2; pp. 834 - 846
• Main Authors: Frederico, Gastão S.F., Torres, Delfim F.M.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15.10.2007; Elsevier
• Subjects: Calculus of variations; Calculus of variations and optimal control; Exact sciences and technology; Fractional derivatives; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Noether's theorem; Partial differential equations; Real functions; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Calculus of variations; Fractional derivatives; Noether's theorem; Integer; Necessary condition; Conservation law; Mathematical analysis; Variational calculus; Optimality condition; Application
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2007.01.013

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler–Lagrange obtained in 2002. Here we use the notion of Euler–Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.