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

References

Whittaker, pp. 80–95.
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.CS1 maint: multiple names: authors list (link)

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[1] Whittaker, pp. 80–95.

[2] Izzo and Biscani