Exact solutions of classical central-force problems

In the classical central-force problem of classical mechanics, some potential energy functions $$V(r)$$ 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 $$r = 1/u$$. Then the Binet equation for $$u(\varphi)$$ can be solved numerically for nearly any central force $$F(1/u)$$. However, only a handful of forces result in formulae for $$u$$ in terms of known functions. The solution for $$\varphi$$ can be expressed as an integral over $$u$$



\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 $$F(r) = ar^{n}$$, then $$u$$ can be expressed in terms of circular functions and/or elliptic functions if $$n$$ 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).

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