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Noether’s Theorem and its Inverse of Birkhoffian System in Event Space Based on Herglotz Variational Problem

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Abstract

Herglotz variational principle, in which the functional is defined by a differential equation, generalizes the classical ones defining the functional by an integral. The principle gives a variational principle description of nonconservative systems even when the Lagrangian is independent of time. This paper focuses on studying the Noether’s theorem and its inverse of a Birkhoffian system in event space based on the Herglotz variational problem. Firstly, according to the Herglotz variational principle of a Birkhoffian system, the principle of a Birkhoffian system in event space is established. Secondly, its parametric equations and two basic formulae for the variation of Pfaff-Herglotz action of a Birkhoffian system in event space are obtained. Furthermore, the definition and criteria of Noether symmetry of the Birkhoffian system in event space based on the Herglotz variational problem are given. Then, according to the relationship between the Noether symmetry and conserved quantity, the Noether’s theorem is derived. Under classical conditions, Noether’s theorem of a Birkhoffian system in event space based on the Herglotz variational problem reduces to the classical ones. In addition, Noether’s inverse theorem of the Birkhoffian system in event space based on the Herglotz variational problem is also obtained. In the end of the paper, an example is given to illustrate the application of the results.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11272227 and 11572212), the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province (No. KYZZ16_0479), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology (No. SKCX16_058).

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Authors and Affiliations

  1. College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, People’s Republic of China

    X. Tian

  2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215011, People’s Republic of China

    Y. Zhang

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  1. X. Tian
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  2. Y. Zhang
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Correspondence to Y. Zhang.

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Tian, X., Zhang, Y. Noether’s Theorem and its Inverse of Birkhoffian System in Event Space Based on Herglotz Variational Problem. Int J Theor Phys 57, 887–897 (2018). https://doi.org/10.1007/s10773-017-3621-2

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  • DOI: https://doi.org/10.1007/s10773-017-3621-2

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