Articles written in Resonance – Journal of Science Education
Volume 25 Issue 4 April 2020 pp 513-538 General Article
The Central Force Problem in n Dimensions
V Balakrishnan Suresh Govindarajan S Lakshmibala
The motion of a particle moving under the influence of acentral force is a fundamental paradigm in dynamics. Theproblem of planetary motion, specifically the derivation ofKepler’s laws motivated Newton’s monumental work, PrincipiaMathematica, effectively signalling the start of modernphysics. Today, the central force problem stands as a basiclesson in dynamics. In this article, we discuss the classicalcentral force problem in a general number of spatialdimensions n, as an instructive illustration of important aspectssuch as integrability, super-integrability and dynamicalsymmetry. The investigation is also in line with the realisationthat it is useful to treat the number of dimensions as avariable parameter in physical problems. The dependenceof various quantities on the spatial dimensionality leads to aproper 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 integrabilityof the central force problem. We then consider an importantsuper-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 maximallysuper-integrable. We discuss the significance of theseCOMs as generators of the dynamical symmetry group of theHamiltonian. This group, the rotation group in n + 1 dimensions,is larger than the kinematical symmetry group for ageneral central force, namely, the rotation group in n dimensions.
Volume 28 | Issue 5