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An Introduction to the Classical Three-Body Problem

From Periodic Solutions to Instabilities and Chaos

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Abstract

The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon’s Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of ‘chaos’. We survey the three-body problem in its historical context and use it to introduce several ideas and techniques that have been developed to understand classical mechanical systems.

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References

  1. J Laskar, Is the Solar SystemStable?, Progress inMathematical Physics, Vol.66, pp.239–270, 2013.

    Article  Google Scholar 

  2. M C Gutzwiller, Chaos in Classical and Quantum mechanics, Springer-Verlag, New York, 1990.

    Book  Google Scholar 

  3. S Bodenmann, The 18th-century Battle Over Lunar Motion, Physics Today, Vol.63, No.1, p.27, 2010.

    Article  Google Scholar 

  4. H Goldstein, C P Poole and J L Safko, Classical Mechanics, 3rd Ed., Pearson Education, 2011.

    Google Scholar 

  5. L N Hand and J D Finch, Analytical Mechanics, Cambridge Univ. Press, 1998.

    Book  Google Scholar 

  6. S G Rajeev, Advanced Mechanics: From Euler’s Determinism to Arnold’s Chaos, Oxford University Press, Oxford, 2013.

    Book  Google Scholar 

  7. F Diacu and P Holmes, Celestial Encounters: The Origins of Chaos and Stability, Princeton University Press, New Jersey, 1996.

    Google Scholar 

  8. Z EMusielak and B Quarles, The Three-body Problem, Reports on Progress in Physics, Vol.77, No.6, p.065901, 2014, arXiv:1508.02312.

    Google Scholar 

  9. J Barrow-Green, Poincare and the Three Body Problem, Amer. Math. Soc., Providence, Rhode Island, 1997.

    Google Scholar 

  10. E T Whittaker, A Treatise on the Analytical Dynamics of Particles & Rigid Bodies, 2nd Ed., Cambridge University Press, Cambridge, 1917, Chapt. XIV and p.283.

    Google Scholar 

  11. K R Symon, Mechanics, 3rd Ed., Addison Wesley, Philippines 1971.

    Google Scholar 

  12. N Mukunda, Sir William Rowan Hamilton, Resonance, 21, No.6, p.493, 2016.

    Book  Google Scholar 

  13. M C Gutzwiller, Moon-Earth-Sun: The Oldest Three-body Problem, Reviews of Modern Physics, Vol.70, p.589, 1998.

    Google Scholar 

  14. D G Saari and Z Xia, Off to Infinity in Finite Time, Notices of the AMS, Vol.42, p.538, 1993

    Google Scholar 

  15. C L Siegel and J K Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, p.31, 1971.

    Book  Google Scholar 

  16. WR Brown, Hypervelocity Stars in theMilkyWay, Physics Today, Vol.69, No.6, p.52, 2016.

    Article  Google Scholar 

  17. R Montgomery, A New Solution to the Three-body Problem, Notices of the AMS, Vol.48, No.5, p.471, 2001.

    Google Scholar 

  18. C Lanczos, The Variational Principles of Mechanics, 4th Ed., Dover, New York, p.139, 1970.

    Google Scholar 

  19. G S Krishnaswami and H Senapati, Curvature and Geodesic Instabilities in a Geometrical Approach to the Planar Three-body Problem, J. Math. Phys., Vol.57, p.102901, 2016, arXiv:1606.05091.

    Article  Google Scholar 

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Correspondence to Govind S Krishnaswami.

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(Left) Govind Krishnaswami is on the faculty of the Chennai Mathematical Institute. His research concerns various problems in theoretical and mathematical physics.

(Right) Himalaya Senapati is a PhD student at the Chennai Mathematical Institute. He works on dynamical systems and chaos.

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Krishnaswami, G.S., Senapati, H. An Introduction to the Classical Three-Body Problem. Reson 24, 87–114 (2019). https://doi.org/10.1007/s12045-019-0760-1

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  • DOI: https://doi.org/10.1007/s12045-019-0760-1

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