Show that the motion of a particle in the potential field is the same as that of the motio...

Show that the motion of a particle in the potential field

is the same as that of the motion under the Kepler potential alone when expressed in terms of a coordinate system rotating or precessing around the center of force.

For negative total energy, show that if the additional potential term is very small compared to the Kepler potential, then the angular speed of precession of the elliptical orbit is

The perihelion of Mercury is observed to precess (after collection for known planetary perturbations) at the rate of about 40″ of arc per century. Show that this precession could be accounted for classically if the dimensionless quantity

(which is a measure of the perturbing inverse-square potential relative to the gravitational potential) were as small as 7 × 10−8. (The eccentricity of Mercury’s orbit is 0.206, and its period is 0.24 year.)