Two-point characteristic function for the Kepler–Coulomb problem

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Abstract
Hamilton’s two‐point characteristic function S (q2t2,q1t1) designates the extremum value of the action integral between two space–time points. It is thus a solution of the Hamilton–Jacobi equation in two sets of variables which fulfils the interchange condition S (q1t1,q2t2) =−S (q2t2,q1t1). Such functions can be used in the construction of quantum‐mechanical Green’s functions. For the Kepler–Coulomb problem, rotational invariance implies that the characteristic function depends on three configuration variables, say r1,r2,r12. The existence of an extra constant of the motion, the Runge–Lenz vector, allows a reduction to two independent variables: x≡r1+r2+r12 and y≡r1+r2−r12. A further reduction is made possible by virtue of a scale symmetry connected with Kepler’s third law. The resulting equations are solved by a double Legendre transformation to yield the Kepler–Coulomb characteristic function in implicit functional form. The periodicity of the characteristic function for elliptical orbits can be applied in a novel derivation of Lambert’s theorem