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The Noether–Bessel-Hagen symmetry approach for dynamical systems

Corresponding author.

Department of Mathematics, Faculty of Civil Engineering, VŠB-Technical University of Ostrava, Ludvíka Podéště 1875/17, 708 00 Ostrava-Poruba, Czech Republic

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Francesco Bajardi

Dipartimento di Fisica “Ettore Pancini”, Universitá degli Studi di Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, 80126 Napoli, Italy

INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy

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and

Salvatore Capozziello

Dipartimento di Fisica “Ettore Pancini”, Universitá degli Studi di Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, 80126 Napoli, Italy

INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy

Gran Sasso Science Institute, Viale Francesco Crispi 7, I-67100 L’Aquila, Italy

Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia

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https://doi.org/10.1142/S0219887820502151Cited by:5

Abstract

The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential $V (φ)$ . We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

Keywords:

AMSC: 70H33, 58E30, 58D19, 53A15

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Published: 20 October 2020

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Received 26 August 2020

Accepted 30 September 2020

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.

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The Noether–Bessel-Hagen symmetry approach for dynamical systems

Abstract

References

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