Exact solutions of classical central-force problems

In the classical central-force problem of classical mechanics, some potential energy functions formula_1 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 formula_2. Then the Binet equation for formula_3 can be solved numerically for nearly any central force formula_4. However, only a handful of forces result in formulae for formula_5 in terms of known functions. The solution for formula_6 can be expressed as an integral over formula_5
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 formula_9, then formula_5 can be expressed in terms of circular functions and/or elliptic functions if formula_11 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 formula_12, the problem also is solved explicitly in terms of Weierstrass elliptic functions.