The Central Force Problem in n Dimensions

Abstract

The motion of a particle moving under the influence of a central force is a fundamental paradigm in dynamics. The problem of planetary motion, specifically the derivation of Kepler’s laws motivated Newton’s monumental work, Principia Mathematica, effectively signalling the start of modern physics. Today, the central force problem stands as a basic lesson in dynamics. In this article, we discuss the classical central force problem in a general number of spatial dimensions n, as an instructive illustration of important aspects such as integrability, super-integrability and dynamical symmetry. The investigation is also in line with the realisation that it is useful to treat the number of dimensions as a variable parameter in physical problems. The dependence of various quantities on the spatial dimensionality leads to a proper perspective of the problems concerned. We consider, first, the orbital angular momentum (AM) in n dimensions, and discuss in some detail the role it plays in the integrability of the central force problem. We then consider an important super-integrable case, the Kepler problem, in n dimensions. The existence of an additional vector constant of the motion (COM) over and above the AM makes this problem maximally super-integrable. We discuss the significance of these COMs as generators of the dynamical symmetry group of the Hamiltonian. This group, the rotation group in n + 1 dimensions, is larger than the kinematical symmetry group for a general central force, namely, the rotation group in n dimensions.

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Acknowledgement

We are grateful to the reviewer for a careful reading of the manuscript and for a number of valuable suggestions for its improvement.

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Correspondence to V. Balakrishnan.

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The authors are with the Department of Physics, IIT Madras, Chennai. Their current research interests are: V Balakrishnan (Stochastic processes and dynamical systems) Suresh Govindarajan (String theory, black holes, statistical physics) S Lakshmibala (Classical and quantum dynamics, quantum optics)

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Balakrishnan, V., Govindarajan, S. & Lakshmibala, S. The Central Force Problem in n Dimensions. Reson 25, 513–538 (2020). https://doi.org/10.1007/s12045-020-0968-0

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Keywords

  • Hamiltonian
  • central force
  • integrability
  • dynamical symmetry
  • constant of the motion
  • Kepler problem
  • algebraically independent constants