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

Jacky Cresson (1) , Anna Szafrańska

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in [10] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [7]) using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular ([9],Theorem 32). However, the counterexample does not explain why and where the proof given in [10] does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in [9] 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 [10]. 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 [9] and obtaining an alternative proof of the main result of Atanackovic and al. [3]. MSC 2010 : Primary 26A33; Secondary 34A08, 70H03.

Mots clés: and Phrases: Euler-Lagrange equations; Noether's theorem; fractional calculus; symmetries

Domaines: Systèmes dynamiques [math.DS]

Jacky Cresson, Anna Szafrańska. About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof. Fractional Calculus and Applied Analysis, 2019, 22 (4), pp.871-898. ⟨10.1515/fca-2019-0048⟩. ⟨hal-02420517⟩