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Orbital Dynamics, Chaotic Orbits and Jacobi Elliptic Functions

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Abstract

Bertrand theorem’s states that, among central-force potentials with bound orbits, there are only two types of central-force scalar potentials with the property that all bound orbits are also closed orbits: the inverse-square law and Hooke’s law. These solutions are considered basic examples in classical mechanics since they help in understanding the regular and predictable motion of bodies and superintegrable dynamical systems. However, there are strong beliefs that other potentials may arise in dynamical systems which are not predicted by Bertrand’s theorem. Besides, several dynamical systems such as the solar system are characterized by chaotic and unbounded orbits which are not predicted by Bertrand’s theorem. In this work, we prove an extension of Bertrand’s theorem by means of non-standard Lagrangians and show the existence of a family of solutions for chaotic unstable periodic orbits.

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Acknowledgements

The authors are indebted for the group of anonymous referees for their useful comments and valuable suggestions.

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The authors would like to thank Chiang Mai University for funding this research.

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Authors and Affiliations

  1. Center of Excellence in Quantum Technology, Faculty of Engineering, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  2. Quantum-Atom Optics Laboratory and Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  3. Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Waranont Anukool

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  1. Rami Ahmad El-Nabulsi
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El-Nabulsi, R.A., Anukool, W. Orbital Dynamics, Chaotic Orbits and Jacobi Elliptic Functions. J Astronaut Sci 70, 1 (2023). https://doi.org/10.1007/s40295-023-00367-x

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