A non-Newtonian Noether's symmetry theorem

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 with 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.

Keywords:

• Non-Newtonian calculus of variations
• Euler–Lagrange extremals
• DuBois–Reymond condition
• Erdmann condition
• symmetry Noether's theorem

2020 Mathematics Subject Classifications:

• 26A24
• 49K05
• 49S05

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by Fundação para a Ciência e a Tecnologia (FCT) and the Center for Research and Development in Mathematics and Applications (CIDMA), project UIDB/04106/2020.