The three-dimensional Coulomb–Kepler problem is shown to be equivalent to a pair of coupled two-dimensional harmonic oscillators with the same angular momentum in classical mechanics by means of a Kustaanheimo–Stiefel transformation of coordinates and velocities. The constraint condition inherent in the transformation is shown to be related to the Runge–Lenz vector. The relationship between the action variables of the two systems is discussed. The equivalence is seen to result in the separability of the problem in parabolic coordinates.
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Augustine C. Chen (1987) studied this question.
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