Exact solutions of classical central-force problems

In the classical central-force problem of classical mechanics, some potential energy functions ${\displaystyle V(r)}$ produce motions or orbits that can be expressed in terms of well-known functions, such as the trigonometric functions and elliptic functions. This article describes these functions and the corresponding solutions for the orbits.

General problem

Let ${\displaystyle r=1/u}$. Then the Binet equation for ${\displaystyle u(\varphi )}$ can be solved numerically for nearly any central force ${\displaystyle F(1/u)}$. However, only a handful of forces result in formulae for ${\displaystyle u}$ in terms of known functions. The solution for ${\displaystyle \varphi }$ can be expressed as an integral over ${\displaystyle u}$

${\displaystyle \varphi =\varphi _{0}+{\frac {L}{\sqrt {2m}}}\int ^{u}{\frac {du}{\sqrt {E_{\mathrm {tot} }-V(1/u)-{\frac {L^{2}u^{2}}{2m}}}}}}$

A central-force problem is said to be "integrable" if this integration can be solved in terms of known functions.

If the force is a power law, i.e., if ${\displaystyle F(r)=ar^{n}}$, then ${\displaystyle u}$ can be expressed in terms of circular functions and/or elliptic functions if ${\displaystyle n}$ equals 1, -2, -3 (circular functions) and -7, -5, -4, 0, 3, 5, -3/2, -5/2, -1/3, -5/3 and -7/3 (elliptic functions).^[1]

If the force is the sum of an inverse quadratic law and a linear term, i.e., if ${\displaystyle F(r)={\frac {a}{r^{2}}}+cr}$, the problem also is solved explicitly in terms of Weierstrass elliptic functions.^[2]