About the Noether’s Theorem for Fractional Lagrangian Systems and a Generalization of the Classical Jost Method of Proof

Abstract

Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in [8] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [6]) using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular ([7],Theorem 32). However, the counterexample does not explain why and where the proof given in [8] does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in [7] 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 [8]. 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 [7] and obtaining an alternative proof of the main result of Atanackovic and al. [2].

Noether’s-type theorems on time scales

“…The terminology of Jost's method was introduced in [12] to designate a particular way of proving the classical Noether's theorem which can be found in [23]. The idea is very simple and elegant.…”

“…The idea being then to apply the well known Noether's theorem in this case to obtain the desired constant of motion. In [12], we have identified several steps in the method:…”

“…Torres in [3] which refers to a time scales analogue of a classical proof exposed by J. Jost and X. Li-Jost in [23]. We point out several difficulties which are in fact inherent to the Jost's method (see [12]). This first proof is not the simplest one but it explains where and why the initial proof given in [3] is not correct.…”

“…• First, they proposed in [3] to derive the Noether's theorem for transformations depending on time from the easier result obtained for transformations without changing the time. In [12], we call Jost's method this way of proving the Noether's theorem as a classical reference is contained in the book [23]. • Another method is proposed in [4], where a second Euler-Lagrange equation is derived ([4], Theorem 5 p.12) and from which the Noether's theorem is deduced (see [4],Section 4, Theorem 6).…”

Fractional Birkhoffian Mechanics Based on Quasi-Fractional Dynamics Models and Its Noether Symmetry

“…Frederico and Torres [41] first presented fractional Noether's theorems. Since then, studies on fractional Noether symmetry and conservation laws have been extensively developed [42][43][44][45][46][47][48][49]. In addition, Torres and Frederico studied Noether's theorems of fractional action-like variation problems [50,51].…”