# 克卜勒問題

${\displaystyle \mathbf {F} ={\frac {k}{r^{2}}}\mathbf {\hat {r}} \,\!}$

${\displaystyle V(r)=-{\frac {k}{r}}\,\!}$

## 克卜勒問題解析

${\displaystyle m{\frac {d^{2}r}{dt^{2}}}-mr\omega ^{2}=m{\frac {d^{2}r}{dt^{2}}}-{\frac {L^{2}}{mr^{3}}}=-{\frac {dV}{dr}}\,\!}$

${\displaystyle {\frac {d}{dt}}={\frac {L}{mr^{2}}}{\frac {d}{d\theta }}\,\!}$

${\displaystyle {\frac {L}{r^{2}}}{\frac {d}{d\theta }}\left({\frac {L}{mr^{2}}}{\frac {dr}{d\theta }}\right)-{\frac {L^{2}}{mr^{3}}}=-{\frac {dV}{dr}}\,\!}$

${\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=-{\frac {m}{L^{2}}}{\frac {d}{du}}V(1/u)\,\!}$

${\displaystyle V(\mathbf {r} )={\frac {-k}{r}}=-ku\,\!}$

${\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=-{\frac {m}{L^{2}}}{\frac {d}{du}}V(1/u)={\frac {km}{L^{2}}}\,\!}$

${\displaystyle u\equiv {\frac {1}{r}}={\frac {km}{L^{2}}}\left[1+e\cos \left(\theta -\theta _{0}\right)\right]\,\!}$

${\displaystyle e={\sqrt {1+{\frac {2EL^{2}}{k^{2}m}}}}\,\!}$

## 參考文獻

1. ^ Arnold, VI. Mathematical Methods of Classical Mechanics, 2nd ed.. New York: Springer-Verlag. 1989: 38. ISBN 0-387-96890-3.
2. ^ Goldstein, H. Classical Mechanics 2nd edition. Addison Wesley. 1980.