In the classical central-force problem of classical mechanics, some potential energy functions V ( r ) {\displaystyle 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 {\displaystyle r=1/u} . Then the Binet equation for u ( φ ) {\displaystyle u(\varphi )} can be solved numerically for nearly any central force F ( 1 / u ) {\displaystyle F(1/u)} . However, only a handful of forces result in formulae for u {\displaystyle u} in terms of known functions. The solution for φ {\displaystyle \varphi } can be expressed as an integral over u {\displaystyle u}
φ = φ 0 + L 2 m ∫ u d u E t o t − V ( 1 / u ) − L 2 u 2 2 m {\displaystyle \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 ) = a r n {\displaystyle F(r)=ar^{n}} , then u {\displaystyle u} can be expressed in terms of circular functions and/or elliptic functions if n {\displaystyle 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 + c r {\displaystyle F(r)={\frac {a}{r^{2}}}+cr} , the problem also is solved explicitly in terms of Weierstrass elliptic functions.2
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. {{cite book}}: ISBN / Date incompatibility (help)
- Izzo,D. and Biscani, F. (2014). Exact Solution to the constant radial acceleration problem. Journal of Guidance Control and Dynamic.{{cite book}}: CS1 maint: multiple names: authors list (link)