Abstract
The separation of variables in the Hamilton-Jacobi Equation is the main tool for the integration of a general mechanical system. In this chapter we study all the orthogonal coordinate systems for which the Kepler Problem is separable. There exist in fact four coordinate systems with this property: the spherical, the parabolic, the elliptic and the spheroconical coordinates’). In Part II we will see why these, and only these, coordinate systems have the separability property, but for the moment we only verify, by an explicit calculation, that the method generates the four triplets of first integrals in involution.
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