Second Noether’s theorem with time delay

Abstract

In this paper, we consider variational problems with delayed argument. Simple proofs of the first and second Noether theorems are given. Applying our results to some concrete problems we obtain conservation laws for systems with delay.

Keywords:

• Time delay
• Noether’s theorem
• calculus of variations
• constant of motion
• Euler–Lagrange equation
• symmetries

AMS Subject Classifications:

• 49S05
• 49K05

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported under the Bialystok University of Technology [grant number S/WI/1/2016](A. B. Malinowska), the Warsaw School of Economics [grant number KAE/S15/35/15] (T. Odzijewicz).