Centrální silové pole — pokračování

F = α (r) \dot{r}

\vec{l} = \vec{r} \times m \vec{v} = \vec{konst.}

m \ddot{r} - m r {\dot{φ}}^{2} = F_{r}

m r \ddot{φ} + 2 m \dot{r} \dot{φ} = 0

\dot{r} = v_{r}

\dot{φ} = ω

l = m r^{2} \dot{φ}

\begin{aligned} E & = \frac{m (v_{r}^{2} + v_{φ}^{2})}{2} + U (r) \\ = \frac{m {\dot{r}}^{2}}{2} + \frac{m r^{2} {\dot{φ}}^{2}}{2} + U (r) \\ = \frac{m {\dot{r}}^{2}}{2} + \underset{U_{e f} (r)}{\underset{⏟}{\frac{l^{2}}{2 m r^{2}} + U (r)}} \end{aligned}

\dot{r} = \pm \sqrt{\frac{2}{m} (E - U_{e f} (r))}

t + t_{0} = \pm \int \frac{d r}{\sqrt{\frac{2}{m} (E - U_{e f} (r))}}

\frac{d φ}{d r} = \frac{\frac{l}{m r^{2}}}{\sqrt{\frac{2}{m} (E - U_{e f} (r))}}

φ = \int \frac{\frac{l}{r^{2}}}{\sqrt{2 m (E - U_{e f} (r))}} d t + φ_{0}

U_{g} = - G \frac{m M}{r}

Na povrchu Země:

\begin{aligned} U_{g} (r) & = U_{g} (R_{Z}) + U_{g}^{'} (R_{Z}) (r - R_{Z}) + \dots \\ \approx - G \frac{m M}{R_{Z}} + G \frac{m M}{R_{Z}^{2}} (r - R_{Z}) \\ = - G \frac{m M}{R_{Z}} + m g h \end{aligned}

DB 🚬