Exact solutions of classical central-force problems explained

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} + (L)/(\sqrt{ 2 m) } \int ^ \frac

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).^[1]

If the force is the sum of an inverse quadratic law and a linear term, i.e., if

F(r) = (a)/(r ^{2}) + cr , the problem also is solved explicitly in terms of Weierstrass elliptic functions.^[2]

Bibliography

• Book: Whittaker ET . E. T. Whittaker . 1937 . . 4th . Dover Publications . New York . 978-0-521-35883-5.
• Book: Izzo,D. and Biscani, F. . 2014 . Exact Solution to the constant radial acceleration problem . Journal of Guidance Control and Dynamic.

Notes and References

1. Whittaker, pp. 80 - 95.
2. Izzo and Biscani