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Open AccessArticle

Herglotz’s Variational Problem for Non-Conservative System with Delayed Arguments under Lagrangian Framework and Its Noether’s Theorem

College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
Symmetry 2020, 12(5), 845; https://doi.org/10.3390/sym12050845
Received: 14 March 2020 / Revised: 6 May 2020 / Accepted: 13 May 2020 / Published: 21 May 2020
Because Herglotz’s variational problem achieves the variational representation of non-conservative dynamic processes, its research has attracted wide attention. The aim of this paper is to explore Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Firstly, we derive the non-isochronous variation formulas of Hamilton–Herglotz action containing delayed arguments. Secondly, for the Hamilton–Herglotz action case, we define the Noether symmetry and give the criterion of symmetry. Thirdly, we prove Herglotz type Noether’s theorem for non-conservative system with delayed arguments. As a generalization, Birkhoff’s version and Hamilton’s version for Herglotz type Noether’s theorems are presented. To illustrate the application of our Noether’s theorems, we give two examples of damped oscillators. View Full-Text
Keywords: non-conservative system with delayed arguments; Noether’s theorem; Herglotz generalized variational principle; Lagrangian framework non-conservative system with delayed arguments; Noether’s theorem; Herglotz generalized variational principle; Lagrangian framework
MDPI and ACS Style

Zhang, Y. Herglotz’s Variational Problem for Non-Conservative System with Delayed Arguments under Lagrangian Framework and Its Noether’s Theorem. Symmetry 2020, 12, 845.

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