Abstract
The universe is in motion. Least action is the principle underlying that motion. Let us look upon the revelations of this principle for the most elementary celestial motion, that of two heavenly bodies such as a planet and the sun. The trajectories of the bodies are the final flowering of the mathematical description. The action principle has provided their equations of motion; however the trajectories do not become apparent until the integrals of the equations of motion are found.
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References
Hermann’s invariant may be found in “Extrait d’une lettre de M. Herman à M. Bernoulli, datée de Padoue le 12. Juillet 1710” published in Histoires de L’Academie Royale des Sciences avec les Mémoires de Mathematique et Physique, Paris, 519–523 (1712). [For a brief survey of Hermann’s life and works, see F. Nagel, Historia Mathematica, 18, 36–54 (1991).] For an historical account of the eccentricity invariant see H. Goldstein, Am. J. Phys. 43, 8 (1975) and 44, 11 (1976) which includes P. S. Laplace, Traité de méchanique céleste, Paris, Tome I, Premiere Partie, Livre II, 165ff (1798–1799); W. R. Hamilton, Proc. R. Irish Acad. 3, Appendix No. III, xxxviff (1847).
See Ian Stewart, The Problems of Mathematics, Oxford: Oxford University Press, Chaps. 9 and 17, 1987.
See Jean Sivardiére, “Laplace vectors for the harmonic oscillator,” Am. J. Phys. 57(6), (1989).
A discussion of Newton’s proof may be found in V. I. Arnol’d, Huygens and Barrow, Newton and Hooke, Basel: Birkhauser, 1990.
See Florin N. Diacu, “Painlevé’s Conjecture,” The Mathematical Intelligencer, 15, No. 2 (1993).
Arnold Sommerfeld, Mechanics, New York: Academic Press, 1956; H. Goldstein, Classical Mechanics (2nd Ed.), Reading Mass: Addison Wesley, 472–478, 1980.
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(2004). Classical Mechanics: The Heavens. In: The Shaggy Steed of Physics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21806-9_4
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DOI: https://doi.org/10.1007/978-0-387-21806-9_4
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