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

Abstract

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.

References in corpus (4)

• A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations
• Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
• Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory
• The Stochastic Nature of Complexity Evolution in the Fractional Systems

Cited by in corpus (1)

• Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense