Noether's theorem for fractional variational problems of variable order

TL;DR

This paper establishes a necessary optimality condition of Euler-Lagrange type for fractional variational problems involving derivatives of variable order, and extends Noether's theorem to these problems without transforming the time variable.

Contribution

It introduces a novel version of Noether's theorem applicable to fractional variational problems with variable order derivatives, without changing the independent variable.

Findings

Derived Euler-Lagrange type optimality condition for variable order fractional derivatives.

Extended Noether's theorem to fractional variational problems of variable order.

Provided a framework for symmetry analysis in fractional calculus of variable order.

Abstract

We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether's theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.