# 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

The Binet equation for u(φ) 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 φ 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) = α rn, 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).[1]

If the force is the sum of a inverse quadratic law and a linear term, i.e., if F(r) = α r-2 + c r, the problem also is solved explicitly in terms of Weierstrass elliptic functions[2]

## References

1. ^ Whittaker, pp. 80–95.
2. ^ Izzo and Biscani

## Bibliography

• Whittaker ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed.). New York: Dover Publications. ISBN 978-0-521-35883-5.
• Izzo,D. and Biscani, F. (2014). Exact Solution to the constant radial acceleration problem. Journal of Guidance Control and Dynamic.