Abstract
Posing Kepler’s problem of motion around a fixed “sun” requires the geometric mechanician to choose a metric and a Laplacian. The metric provides the kinetic energy. The fundamental solution to the Laplacian (with delta source at the “sun”) provides the potential energy. Posing Kepler’s three laws (with input from Galileo) requires symmetry conditions. The metric space must be homogeneous, isotropic, and admit dilations. Any Riemannian manifold enjoying these three symmetry properties is Euclidean. So if we want a semblance of Kepler’s three laws to hold but also want to leave the Euclidean realm, we are forced out of the realm of Riemannian geometries. The Heisenberg group (a subRiemannian geometry) and lattices provide the simplest examples of metric spaces enjoying a semblance of all three of the Keplerian symmetries. We report success in posing, and solving, the Kepler problem on the Heisenberg group. We report failures in posing the Kepler problem on the rank two lattice and partial success in solving the problem on the integers. We pose a number of questions.
For Jerry
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin-Cummings, Reading (1978)
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Springer, Berlin (2010)
Albouy, A.: Projective dynamics and classical gravitation. arXiv:math-ph/0501026v2 (2005)
Cserti, J.: Application of the lattice Green’s function for calculating the resistance of infinite networks of resistors. Am. J. Phys. 68, 896–906. arXiv:cond-mat/9909120v4 (2000).
Folland, G.: A fundamental solution for a subelliptic operator. Bull. Am. Math. Soc. 79, 373–376 (1973)
Diacu, F., Perez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. Part I: relative equilibria. J. Nonlinear Sci. 22, 247–266 (2012)
Diacu, F., Perez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. Part II: singularities. J. Nonlinear Sci. 22, 267–275 (2012)
Diacu, F., Perez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. arXiv:0807.1747v6 [math.DS] (2008)
Diacu, F.: The non-existence of centre-of-mass and linear-momentum integrals in the curved N-body problem. arXiv:1202.4739v1 [math.DS] (2012)
Darboux, G.: Étude d’une question relative au mouvement d’un point sur une surface de révolution. Bull. Soc. Math. Franc. 5, 100–113 (1887)
Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. IHES 53, 53–73 (1981)
Gromov, M.: Metric Structures for Riemannian and Non-riemannian Spaces, 3rd printing. Birkhauser, Boston (2007)
Hollos, S., Hollos, R.: The lattice Green function for the Poisson equation on an infinite square lattice. arXiv:cond-mat/0509002v1 [cond-mat.other] (2005)
Lobachevsky, N.I.: The new foundations of geometry with full theory of parallels [in Russian], 1835–1838. In: Collected Works, V. 2, GITTL, Moscow, p. 159 (1949)
Montgomery, R.: A tour of subRiemannian geometries. In: Mathematical Surveys and Monographs, vol. 91. American Mathematical Society, Providence (2002). http://www.ams.org/books/surv/091/ and http://www.ams.org/books/surv/091/surv091-endmatter.pdf
Santoprete, M.: Gravitational and harmonic oscillator potentials on surfaces of revolution. J. Math. Phys. 49, 042903, 16 pp (2008)
Serret, P.J.: Théorie nouvelle géométrique et mécanique des lignes a double courbure. Librave de Mallet-Bachelier, Paris (1860)
Shanbrom, C.: Two problems in sub-Riemannian geometry. Ph.D. Thesis, UC Santa Cruz (2013)
Shchepetilov, A.: Nonintegrability of the two-body problem in constant curvature spaces. J. Phys. A 39, 5787–5806 (2006)
Zagryadskii, O.A., Kudryavtseva, E.A., Fedoseev, D.A.: A generalization of Bertrand’s theorem to surfaces of revolution. Sbornik 203, 39–78 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
Montgomery, R., Shanbrom, C. (2015). Keplerian Dynamics on the Heisenberg Group and Elsewhere. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_14
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2441-7_14
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2440-0
Online ISBN: 978-1-4939-2441-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)