In the classical central-force problem of classical mechanics, some potential energy functions 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[edit]
Let . Then the Binet equation for can be solved numerically for nearly any central force . However, only a handful of forces result in formulae for in terms of known functions. The solution for can be expressed as an integral over
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 , then can be expressed in terms of circular functions and/or elliptic functions if 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 , the problem also is solved explicitly in terms of Weierstrass elliptic functions.[2]
References[edit]
Bibliography[edit]
- 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.
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