A continuous/discrete fractional Noether's theorem

TL;DR

This paper establishes a fractional Noether's theorem applicable to both continuous and discrete fractional Lagrangian systems, providing explicit, computable conservation laws that extend classical symmetry principles to fractional calculus.

Contribution

It introduces a novel fractional Noether's theorem for continuous and discrete systems, with explicit formulas and algorithmic implementation for conservation laws.

Findings

Explicit conservation laws derived for fractional systems

Algorithmic implementation of conservation laws in discrete case

Finite-step computability of conservation laws in discrete systems

Abstract

We prove a fractional Noether's theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. This provides an explicit conservation law (first integral) given by a closed formula which can be algorithmically implemented. In the discrete case, the conservation law is moreover computable in a finite number of steps.