Separation of Variables and Action-Angle Coordinates

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.

Taking the three integrals as new momenta in the canonical formalism, the Hamilton-Jacobi method then gives, after three integrations, the conjugate coordinates. Instead of the integrals directly generated by the separation process, one can take as new momenta a combination of them, and in particular the so-called action coordinates. It turns out that the corresponding conjugate coordinates are angle variables. The action-angle variables are, in some sense, very natural for the topology of the integrable systems, as elucidated by the Arnold Theorem. We will find them for the first two of the above coordinate systems: from the spherical coordinates we will obtain the well known Delaunay variables, and from the parabolic coordinates the Pauli variables. As will become