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Minimizers for the Kepler Problem

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Abstract

We characterize the minimizing geodesics for the Kepler problem endowed with the Jacobi-Maupertuis metric. We focus on the positive energy case, but do all energies. The more complicated negative energy case was solved in Jacobi (Crelles J 17:68–82, 1837. https://doi.org/10.1515/crll.1837.17.68), with his work translated and completed by Todhunter (Researches in the Calculus of Variations, Principally on the Theory of Discontinuous Solutions. Macmillan and Co., Cambridge, 1871), and later summarized in Wintner’s book. Our discussion of these old results includes a new proof for the positive energy case and perspectives coming from metric and differential geometry. For the negative energy result we need Lambert’s theorem which we discuss.

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References

  1. Albouy, A.: Lambert’s Theorem: Geometry or Dynamics? arXiv:1711.03049

  2. Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI, (2001)

  3. Jacobi, C.G.: Zur Theorie der Variations-Rechnung und der Differential-Gleichungen, ournal für die reine und angewandte Mathematik. Crelles J. 17, 68–82 (1837). https://doi.org/10.1515/crll.1837.17.68

    Article  Google Scholar 

  4. Maderna, E., Venturelli, A.: Viscosity Solutions and Hyperbolic Motions: A New PDE method for the N-body problem arXiv:1908.09252

  5. Moeckel, R., Montgomery, R., Sanchez, H.: Free time minimizers for the planar three-body problem, Celestial Mechanics and Dynamical Astronomy, Vol. 130, no. 3, (2018)

  6. Moeckel, R., Montgomery, R., Venturelli, A.: From brake to syzygy. Arch. Rat. Mech. Anal. 204(3), 1009–1060 (2012)

    Article  MathSciNet  Google Scholar 

  7. Montgomery, R.: The Kepler Cone and Weak KAM. preprint, available by email request

  8. Pollard, H.: Celestial Mechanics , MAA, (1977)

  9. Todhunter, M.A.: Researches in the Calculus of Variations, Principally on the Theory of Discontinuous Solutions. Macmillan and Co., Cambridge (1871)

    MATH  Google Scholar 

  10. Todhunter, M.A.: History of the Calculus of Variations. Macmillan and Co., Cambridge (1871)

    MATH  Google Scholar 

  11. Wintner, A.: Analytical Foundations of Celestial Mechanics. Princeton Mathematical Series, vol. 5. Princeton U. Press, Princeton (1947)

    MATH  Google Scholar 

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Acknowledgements

Thank you to Alain Albouy for his clear writings on Lambert’s theorem, for many useful email discussions, and pointers to the literature. Thank you to Hector Sanchez and Rick Moeckel for ongoing email discussions and interest. Thank you to Andrea Venturelli and Ezequiel Maderna for catching a basic error in the statement of Corollary 3.1.

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  1. Mathematics Department, University of California, Santa Cruz, CA, 95064, USA

    Richard Montgomery

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  1. Richard Montgomery
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Correspondence to Richard Montgomery.

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Montgomery, R. Minimizers for the Kepler Problem . Qual. Theory Dyn. Syst. 19, 31 (2020). https://doi.org/10.1007/s12346-020-00363-8

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