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A generalized Noether theorem for scaling symmetry

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Abstract

The recently discovered conserved quantity associated with Kepler rescaling is generalized by an extension of Noether’s theorem which involves the classical action integral as an additional term. For a free particle, the familiar Schrödinger dilations are recovered. A general pattern arises for homogeneous potentials. The associated conserved quantity allows us to derive the virial theorem. The relation to the Bargmann framework is explained and illustrated by exact plane gravitational waves.

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Notes

  1. (5.8) is only a pp wave, not a vacuum solution of the Einstein equations [6, 21].

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Acknowledgements

We are indebted to Gary Gibbons for advice and Bruno Nachtergaele for correspondence. ME thanks the Denis Poisson Institute of Orléans - Tours University for a post-doctoral scholarship. This work was partially supported by National Natural Science Foundation of China (Grant No. 11575254). P.K. was supported by the Grant 2016/23/B/ST2/00727 of the National Science Centre of Poland.

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Authors and Affiliations

  1. School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, 519082, China

    P.-M. Zhang

  2. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China

    P.-M. Zhang & M. Elbistan

  3. Institut Denis Poisson, Tours University – Orléans University, UMR 7013, Tours, France

    M. Elbistan & P. A. Horvathy

  4. Department of Computer Science, Faculty of Physics and Applied Informatics, University of Lódź, Pomorska 149/153, 90-236, Lódź, Poland

    P. Kosiński

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  1. P.-M. Zhang
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  4. P. Kosiński
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Correspondence to M. Elbistan.

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Zhang, PM., Elbistan, M., Horvathy, P.A. et al. A generalized Noether theorem for scaling symmetry. Eur. Phys. J. Plus 135, 223 (2020). https://doi.org/10.1140/epjp/s13360-020-00247-5

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