An interesting class of exact solutions to Binet's equation, with references to Newton's and Einstein's theories of gravity

Abstract

Abstract To adequately describe the motion of a particle in a central force field, one has to specify the apsidal angle (α). The particle's trajectory can only be truly periodic (closed), if α/π is a rational number. For example, F∼1/r^2 then α=π, and the trajectory is an ellipse. This is the simplest case when the trajectory equation (Binet's equation) is exactly solvable. In this article, bounded motion in a two-component central potential field V(r)=-a_1/r-a_2/r^2 will be examined and illustrated. This case is more complex, but the corresponding Binet's equation is still exactly solvable. Moreover, the perturbational solution to the relativistic trajectory equation is, in fact, the exact solution for a force field of the same type. It means that the relativistic force F∼1/r^4 is formally replaced with an effective F∼1/r^3 force. As will be shown, this differently interpretable solution gives a very accurate prediction to the anomalous shift of Mercury's perihelion.