Variational problems with fractional derivatives: Invariance conditions and N\"{o}ther's theorem

TL;DR

This paper extends variational principles and N"{o}ther's theorem to fractional derivatives, establishing invariance conditions and conservation laws for fractional Euler-Lagrangian equations with illustrative examples.

Contribution

It introduces a generalized variational principle for fractional derivatives, derives invariance conditions, and extends N"{o}ther's theorem to fractional calculus with practical approximation methods.

Findings

Derived necessary and sufficient invariance conditions for fractional variational problems.

Extended N"{o}ther's theorem to fractional derivatives, leading to new conservation laws.

Provided concrete examples illustrating the theoretical results.

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 N\"{o}ther's identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of N\"{o}ther'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.