About the Noether's theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof

TL;DR

This paper analyzes and corrects the proof of a fractional Noether's theorem, extending the classical Jost method to fractional Lagrangian systems and clarifying issues in previous works.

Contribution

It provides a detailed analysis and correction of the fractional Noether's theorem proof based on the Jost method, including its fractional generalization and implications.

Findings

Corrected the proof of fractional Noether's theorem

Identified difficulties in defining variational symmetries in fractional calculus

Provided an alternative proof aligning with previous results

Abstract

Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in Appl. Math. Comp. 217,3,2010 was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in JMAA 429, 2, 2015 using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular(JMAA 334, 2007,Theorem 32). However, the counterexample does not explain why and where the proof given in Appl. Math. Comp. 217,3,2010 does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in JMAA 334, 2007 which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in Appl. Math. Comp. 217,3,2010. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost's method, in particular in the discrete setting. We then derive a fractional Noether's Theorem following this strategy, correcting the initial statement of Frederico and Torres in JMAA 334, 2007 and obtaining an alternative proof of the main result of Atanackovic and al. in Nonlinear Analysis 71, 2009.