חוק ראשון של קפלר

L=mr^{2}\dot{\theta}

E=\frac{1}{2}mv^{2}-\frac{GMm}{r}=\frac{1}{2}m\dot{r}^{2}+\frac{1}{2}mr^{2}\dot{\theta}^{2}-\frac{GMm}{r}=\frac{1}{2}m\dot{r}^{2}+\frac{L^{2}}{2mr^{2}}-\frac{GMm}{r}

r'=\frac{dr}{dt}\cdot\frac{dt}{d\theta}=\frac{\dot{r}}{\dot{\theta}}=\dot{r}\cdot\frac{mr^{2}}{L}

E=\frac{1}{2}mr'^{2}\cdot\left(\frac{L}{mr^{2}}\right)^{2}+\frac{L^{2}}{2mr^{2}}-\frac{GMm}{r}=\frac{L^{2}}{2m}\left(\frac{r'^{2}}{r^{4}}+\frac{1}{r^{2}}-\frac{2GMm^{2}}{L^{2}}\frac{1}{r}\right)

u'=-\frac{1}{r^{2}}r'

\frac{2mE}{L^{2}}=u'^{2}+u^{2}-\frac{2GMm^{2}}{L^{2}}u=u'^{2}+\left(u-\frac{GMm^{2}}{L^{2}}\right)^{2}-\frac{G^{2}M^{2}m^{4}}{L^{4}}

u'^{2}+\left(u-\frac{GMm^{2}}{L^{2}}\right)^{2}=\frac{2mE}{L^{2}}+\frac{G^{2}M^{2}m^{4}}{L^{4}}=\const

u\left(\theta\right)=\sqrt{\left(\frac{2mE}{L^{2}}+\frac{G^{2}M^{2}m^{4}}{L^{4}}\right)}\cos\left(\theta+\phi\right)+\frac{GMm^{2}}{L^{2}}=\frac{GMm^{2}}{L^{2}}\left(\sqrt{1+\frac{2EL^{2}}{G^{2}M^{2}m^{3}}}\cos\left(\theta+\phi\right)+1\right)

r\left(\theta\right)=\frac{1}{u\left(\theta\right)}=\frac{l}{1+\varepsilon\cos\left(\theta+\phi\right)}