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Variational symmetries of Lagrangian systems with second-order derivatives

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Abstract

We discuss an elementary derivation of variational symmetries and corresponding integrals of motion for the Lagrangian systems depending on acceleration. Providing several examples, we make the manuscript accessible to a wide range of readers with an interest in higher-order Lagrangians and symmetries. The discussed technique is also applicable to the Lagrangian systems with higher-order derivatives.

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Notes

  1. It is impossible to cite all relevant papers, we mention here some more modern works for the interested reader [2,3,4,5].

  2. Assuming we have a non-degenerate system, i.e. \(\det \left( \dfrac{\partial ^2\,L}{\partial \ddot{x}_i\partial \ddot{x}_j}\right) \ne 0.\)

  3. In the literature, sometimes this expression is called Noether-Bassel-Hagen identity, see e.g. [14].

  4. Of course, if one knows the symmetries in advance, then it is easier to use the Noether theorem to obtain the integrals of motion.

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Acknowledgements

We would like to thank all the participants of the seminar series on “Higher-derivative systems” held at Bogazici University in the summer of 2020. The work of Ilmar Gahramanov is partially supported by the Bogazici University Research Fund under grant number 20B03SUP3. Ege Çoban and Dilara Kosva are supported by the 2209-A TUBITAK National/International Research Projects Fellowship Programme for Undergraduate Students under grant number 1919B012000987.

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Authors and Affiliations

  1. Department of Physics, Bogazici University, 34342, Bebek, Istanbul, Turkey

    Ege Coban, Ilmar Gahramanov & Dilara Kosva

  2. Institute of Radiation Problems, Azerbaijan National Academy of Sciences, B.Vahabzade St. 9, 1143, Baku, Azerbaijan

    Ilmar Gahramanov

  3. Department of Mathematics, Khazar University, Mehseti St. 41, 1096, Baku, Azerbaijan

    Ilmar Gahramanov

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  1. Ege Coban
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Correspondence to Ilmar Gahramanov.

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Coban, E., Gahramanov, I. & Kosva, D. Variational symmetries of Lagrangian systems with second-order derivatives. Eur. Phys. J. Plus 138, 605 (2023). https://doi.org/10.1140/epjp/s13360-023-04241-5

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