A Non-Newtonian Noether's Symmetry Theorem

TL;DR

This paper extends Noether's symmetry theorem to the non-Newtonian calculus of variations, establishing a conserved quantity for variational problems within this new mathematical framework.

Contribution

It introduces a novel proof of Noether's theorem using a new optimality condition tailored for non-Newtonian calculus.

Findings

Proves Noether's theorem in the context of non-Newtonian calculus

Establishes a new necessary optimality condition of DuBois-Reymond type

Demonstrates the existence of conserved quantities in this framework

Abstract

The universal principle obtained by Emmy Noether in 1918, asserts that the invariance of a variational problem with respect to a one-parameter family of symmetry transformations implies the existence of a conserved quantity along the Euler-Lagrange extremals. Here we prove Noether's theorem for the recent non-Newtonian calculus of variations. The proof is based on a new necessary optimality condition of DuBois-Reymond type.