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 $u$

:$\varphi = \varphi_ + \frac \int ^ \frac$

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 $F(r) = ar^$, 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 $F(r) = \frac + cr$, the problem also is solved explicitly in terms of Weierstrass elliptic functions.Izzo and Biscani

References

Bibliography

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* {{cite book , author = Izzo,D. and Biscani, F. , year = 2014 , title = Exact Solution to the constant radial acceleration problem , publisher = Journal of Guidance Control and Dynamic

Classical mechanics