The Two-Body or Central Field Problem

The central field problem distinguishes celestial mechanics from other areas of classical dynamics. This deals with the motion of a test particle, whose mass is negligible with respect to the central body, in the gravitational field of a point mass. The extended version of this problem is to allow the central mass to have a finite spatial extent, to depart from spherical symmetry, and perhaps to rotate. The basic Newtonian problem is the foUowing. 

The equation of motion for a single particle under the action of an arbitrary force, F, is mr =F, where m is the mass of the particle and r is its position vector. In most of what follows, we will assume a cylindrical or spherical coordinate system (which will be specified), the most appropriate one for use with planar motion. The radial component of the momentum equation is 

Here the solution to the central field problem makes specific use of the gravitational force. The substitution of 

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