Exact Solutions in the Theory of Orbits
Open Access
- 1 December 1966
- journal article
- Published by Oxford University Press (OUP) in Monthly Notices of the Royal Astronomical Society
- Vol. 134 (3) , 253-268
- https://doi.org/10.1093/mnras/134.3.253
Abstract
It is known that the Hamilton–Jacobi and Schrödinger equations can be separated in the eleven coordinate systems for which Laplace's equation is separable, provided the potential is of suitable form. The problem studied in this paper is which of these potentials are also solutions of Laplace's equation. It is found that such solutions exist only for eight of the eleven coordinate systems. The principal result of the paper is that no solution exists in ellipsoidal coordinates and hence that Vinti's (1959) exact solution of the problem of orbits in spheroidal coordinates cannot be extended to ellipsoidal coordinates.Keywords
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