Unterabschnitte

Beschleunigung

Die Beschleunigung ist in kartesischen Koordinaten

$\displaystyle \vec{a}=\frac{d^{2}\vec{r}}{dt^{2}}= \begin{pmatrix}\frac{d^{2}x}...
...t{z} \end{pmatrix} =\ddot{x}\vec{e}_{x}+\ddot{y}\vec{e}_{y}+\ddot{z}\vec{e}_{z}$ (G..575)

Wir verwenden die Beziehungen

$\displaystyle x =$ $\displaystyle r\sin(\theta)\cos(\phi)$ (G..576)
$\displaystyle y =$ $\displaystyle r\sin(\theta)\sin(\phi)$ (G..577)
$\displaystyle z =$ $\displaystyle r\cos(\theta)$ (G..578)

und leiten sie zweimal ab. Wir erhalten aus

$\displaystyle \dot{x} =$ $\displaystyle \dot{r}\sin(\theta)\cos(\phi)+r\cos(\theta)\cos(\phi)\dot{\theta }-r\sin(\theta)\sin(\phi)\dot{\phi}$    
$\displaystyle \dot{y} =$ $\displaystyle \dot{r}\sin(\theta)\sin(\phi)+r\cos(\theta)\sin(\phi)\dot{\theta }+r\sin(\theta)\cos(\phi)\dot{\phi}$    
$\displaystyle \dot{z} =$ $\displaystyle \dot{r}\cos(\theta)-r\sin(\theta)\dot{\theta}$    

die Gleichungen

$\displaystyle \ddot{x}=$ $\displaystyle \ddot{r}\sin(\theta)\cos(\phi)+\dot{r}\cos(\theta)\cos(\phi )\dot{\theta}-\dot{r}\sin(\theta)\sin(\phi)\dot{\phi}$ (G..579)
  $\displaystyle +\dot{r}\cos(\theta)\cos(\phi)\dot{\theta}-r\sin(\theta)\cos(\phi...
...s(\theta)\sin(\phi)\dot{\phi}\dot{\theta}+r\cos (\theta)\cos(\phi)\ddot{\theta}$    
  $\displaystyle -\dot{r}\sin(\theta)\sin(\phi)\dot{\phi}-r\cos(\theta)\sin(\phi)\...
...heta}-r\sin(\theta)\cos(\phi)\dot{\phi}^{2}-r\sin(\theta)\sin (\phi)\ddot{\phi}$    
$\displaystyle =$ $\displaystyle \ddot{r}\sin(\theta)\cos(\phi)$    
  $\displaystyle +\dot{r}\dot{\theta}\left[ \cos(\theta)\cos(\phi)+\cos(\theta)\cos (\phi)\right]$    
  $\displaystyle +\dot{r}\dot{\phi}\left[ -\sin(\theta)\sin(\phi)-\sin(\theta)\sin (\phi)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\phi)\right]$    
  $\displaystyle +r\dot{\phi}\dot{\theta}\left[ -\cos(\theta)\sin(\phi)-\cos(\theta )\sin(\phi)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ \cos(\theta)\cos(\phi)\right]$    
  $\displaystyle +r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\phi)\right]$    
  $\displaystyle +r\ddot{\phi}\left[ -\sin(\theta)\sin(\phi)\right]$    
$\displaystyle =$ $\displaystyle \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos(\theta)\c...
...dot{r}\dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta}^{2}\sin (\theta)\cos(\phi)$    
  $\displaystyle -2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\theta}\co...
...\phi)-r\dot{\phi}^{2}\sin(\theta)\cos(\phi)-r\ddot{\phi} \sin(\theta)\sin(\phi)$    


und

$\displaystyle \ddot{y} =$ $\displaystyle \ddot{r}\sin(\theta)\sin(\phi)+\dot{r}\cos(\theta)\sin(\phi )\dot{\theta}+\dot{r}\sin(\theta)\cos(\phi)\dot{\phi}$ (G..580)
  $\displaystyle +\dot{r}\cos(\theta)\sin(\phi)\dot{\theta}-r\sin(\theta)\sin(\phi...
...s(\theta)\cos(\phi)\dot{\theta}\dot{\phi}+r\cos (\theta)\sin(\phi)\ddot{\theta}$    
  $\displaystyle +\dot{r}\sin(\theta)\cos(\phi)\dot{\phi}+r\cos(\theta)\cos(\phi)\...
...\phi}-r\sin(\theta)\sin(\phi)\dot{\phi}^{2}+r\sin(\theta)\cos(\phi )\ddot{\phi}$    
$\displaystyle =$ $\displaystyle \ddot{r}\sin(\theta)\sin(\phi)$    
  $\displaystyle +\dot{r}\dot{\theta}\left[ \cos(\theta)\sin(\phi)+\cos(\theta)\sin (\phi)\right]$    
  $\displaystyle +\dot{r}\dot{\phi}\left[ \sin(\theta)\cos(\phi)+\sin(\theta)\cos (\phi)\right]$    
  $\displaystyle -r\dot{\theta}^{2}\sin(\theta)\sin(\phi)$    
  $\displaystyle +r\dot{\theta}\dot{\phi}\left[ \cos(\theta)\cos(\phi)+\cos(\theta)\cos (\phi)\right]$    
  $\displaystyle +r\cos(\theta)\sin(\phi)\ddot{\theta}$    
  $\displaystyle -r\dot{\phi}^{2}\sin(\theta)\sin(\phi)$    
  $\displaystyle +r\sin(\theta)\cos(\phi)\ddot{\phi}$    
$\displaystyle =$ $\displaystyle \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\theta)\s...
...dot{r}\dot{\phi}\sin(\theta)\cos(\phi)-r\dot{\theta}^{2}\sin (\theta)\sin(\phi)$    
  $\displaystyle +2r\dot{\theta}\dot{\phi}\cos(\theta)\cos(\phi)+r\ddot{\theta}\co...
...\phi)-r\dot{\phi}^{2}\sin(\theta)\sin(\phi)+r\ddot{\phi} \sin(\theta)\cos(\phi)$    

sowie

$\displaystyle \ddot{z} =$ $\displaystyle \ddot{r}\cos(\theta)-\dot{r}\sin(\theta)\dot{\theta }$ (G..581)
  $\displaystyle -\dot{r}\sin(\theta)\dot{\theta}-r\cos(\theta)\dot{\theta}^{2}-r\sin (\theta)\ddot{\theta}$    
$\displaystyle =$ $\displaystyle \ddot{r}\cos(\theta)-2\dot{r}\sin(\theta)\dot{\theta}-r\cos(\theta )\dot{\theta}-r\sin(\theta)\ddot{\theta}$    

Wir setzen in die Gleichung G.24 die Gleichungen G.8, G.9, G.10, G.28, G.29 und G.30 ein und ordnen nach $ \vec{e}_{r}$, $ \vec{e}_{\theta}$ und $ \vec{e}_{\phi}$.

$\displaystyle \vec{a}=$ $\displaystyle \ddot{x}\vec{e}_{x}+\ddot{y}\vec{e}_{y}+\ddot{z}\vec{e}_{z}$ (G..582)
$\displaystyle =$ $\displaystyle \ddot{x}\left[ \sin(\theta)\cos(\phi)\vec{e}_{r}+\cos(\theta)\cos (\phi)\vec{e}_{\theta}-\sin(\phi)\vec{e}_{\phi}\right]$    
  $\displaystyle +\ddot{y}\left[ \sin(\theta)\sin(\phi)\vec{e}_{r}+\cos(\theta)\sin (\phi)\vec{e}_{\theta}+\cos(\phi)\vec{e}_{\phi}\right]$    
  $\displaystyle +\ddot{z}\left[ \cos(\theta)\vec{e}_{r}-\sin(\theta)\vec{e}_{\theta }\right]$    
$\displaystyle =$ $\displaystyle \left[ \ddot{x}\sin(\theta)\cos(\phi)+\ddot{y}\sin(\theta)\sin(\phi )+\ddot{z}\cos(\theta)\right] \vec{e}_{r}$    
  $\displaystyle +\left[ \ddot{x}\cos(\theta)\cos(\phi)+\ddot{y}\cos(\theta)\sin(\phi )-\ddot{z}\sin(\theta)\right] \vec{e}_{\theta}$    
  $\displaystyle +\left[ -\ddot{x}\sin(\phi)+\ddot{y}\cos(\phi)\right] \vec{e}_{\phi }$    

Der Übersichtlichkeit halber berechnen wir nun die drei Komponenten $ \vec{e}_{r}$, $ \vec{e}_{\theta}$ und $ \vec{e}_{\phi}$ getrennt. Wir beginnen mit $ \vec{e}_{r}$.

$\displaystyle a_{r} =$ $\displaystyle \ddot{x}\sin(\theta)\cos(\phi)+\ddot{y}\sin(\theta)\sin(\phi )+\ddot{z}\cos(\theta)$ (G..583)
$\displaystyle =$ $\displaystyle \left[ \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos(\t...
...dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta}^{2} \sin(\theta)\cos(\phi)\right.$    
  $\displaystyle -2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\theta}\cos (\theta)\cos(\phi)$ (G..584)
  $\displaystyle \left. -r\dot{\phi}^{2}\sin(\theta)\cos(\phi)-r\ddot{\phi}\sin(\theta )\sin(\phi)\right] \sin(\theta)\cos(\phi)$    
  $\displaystyle +\left[ \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\theta )\sin(\phi)+2\dot{r}\dot{\phi}\sin(\theta)\cos(\phi)\right.$    
  $\displaystyle -r\dot{\theta}^{2}\sin(\theta)\sin(\phi)+2r\dot{\theta}\dot{\phi}\cos (\theta)\cos(\phi)$    
  $\displaystyle \left. +r\cos(\theta)\sin(\phi)\ddot{\theta}-r\dot{\phi}^{2}\sin ...
...ta)\sin(\phi)+r\ddot{\phi}\sin(\theta)\cos(\phi)\right] \sin(\theta )\sin(\phi)$    
  $\displaystyle +\left[ \ddot{r}\cos(\theta)-2\dot{r}\dot{\theta}\sin(\theta)-r\dot{\theta }^{2}\cos(\theta)-r\ddot{\theta}\sin(\theta)\right] \cos(\theta)$    
$\displaystyle =$ $\displaystyle \ddot{r}\left[ \sin(\theta)\cos(\phi)\sin(\theta)\cos(\phi)+\sin (\theta)\sin(\phi)\sin(\theta)\sin(\phi)+\cos(\theta)\cos(\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\theta}\left[ \cos(\theta)\cos(\phi)\sin(\theta)\co...
...)+\cos(\theta)\sin(\phi)\sin(\theta)\sin(\phi)-\sin(\theta)\cos (\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\phi}\left[ -\sin(\theta)\sin(\phi)\sin(\theta)\cos (\phi)+\sin(\theta)\cos(\phi)\sin(\theta)\sin(\phi)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\phi)\sin(\theta)\cos(...
...)-\sin(\theta)\sin(\phi)\sin(\theta)\sin(\phi)-\cos(\theta)\cos(\theta )\right]$    
  $\displaystyle +2r\dot{\theta}\dot{\phi}\left[ -\cos(\theta)\sin(\phi)\sin(\theta )\cos(\phi)+\cos(\theta)\cos(\phi)\sin(\theta)\sin(\phi)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ \cos(\theta)\cos(\phi)\sin(\theta)\cos(\phi )+\cos(\theta)\sin(\phi)\sin(\theta)\sin(\phi)-\sin(\theta)\cos(\theta )\right]$    
  $\displaystyle +r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\phi)\sin(\theta)\cos(\phi )-\sin(\theta)\sin(\phi)\sin(\theta)\sin(\phi)\right]$    
  $\displaystyle +r\ddot{\phi}\left[ -\sin(\theta)\sin(\phi)\sin(\theta)\cos(\phi )+\sin(\theta)\cos(\phi)\sin(\theta)\sin(\phi)\right]$    
$\displaystyle =$ $\displaystyle \ddot{r}\left[ \sin^{2}(\theta)\cos^{2}(\phi)+\sin^{2}(\theta)\sin ^{2}(\phi)+\cos^{2}(\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\theta}\left[ \cos(\theta)\sin(\theta)\cos^{2}(\phi )+\cos(\theta)\sin(\theta)\sin^{2}(\phi)-\sin(\theta)\cos(\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\phi}\left[ -\sin^{2}(\theta)\sin(\phi)\cos(\phi)+\sin ^{2}(\theta)\cos(\phi)\sin(\phi)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ -\sin^{2}(\theta)\cos^{2}(\phi)-\sin^{2} (\theta)\sin^{2}(\phi)-\cos^{2}(\theta)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ \cos(\theta)\sin(\theta)\cos^{2}(\phi)+\cos (\theta)\sin(\theta)\sin^{2}(\phi)-\sin(\theta)\cos(\theta)\right]$    
  $\displaystyle +r\dot{\phi}^{2}\left[ -\sin^{2}(\theta)\cos^{2}(\phi)-\sin^{2}(\theta )\sin^{2}(\phi)\right]$    
  $\displaystyle +r\ddot{\phi}\left[ -\sin^{2}(\theta)\sin(\phi)\cos(\phi)+\sin^{2} (\theta)\cos(\phi)\sin(\phi)\right]$    
$\displaystyle =$ $\displaystyle \ddot{r}\left[ \sin^{2}(\theta)+\cos^{2}(\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\theta}\left[ \cos(\theta)\sin(\theta)-\sin(\theta )\cos(\theta)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ -\sin^{2}(\theta)-\cos^{2}(\theta)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ \cos(\theta)\sin(\theta)-\sin(\theta)\cos (\theta)\right]$    
  $\displaystyle +r\dot{\phi}^{2}\left[ -\sin^{2}(\theta)\right]$    
$\displaystyle =$ $\displaystyle \ddot{r}-r\dot{\theta}^{2}-r\sin^{2}(\theta)\dot{\phi}^{2}$    

und

$\displaystyle a_{\theta} =$ $\displaystyle \ddot{x}\cos(\theta)\cos(\phi)+\ddot{y}\cos(\theta)\sin (\phi)-\ddot{z}\sin(\theta)$ (G..585)
$\displaystyle =$ $\displaystyle \left[ \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos(\t...
...dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta}^{2} \sin(\theta)\cos(\phi)\right.$    
  $\displaystyle \left. -2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\th...
...ta)\cos(\phi)-r\ddot{\phi} \sin(\theta)\sin(\phi)\right] \cos(\theta)\cos(\phi)$    
  $\displaystyle +\left[ \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\...
...dot{\phi}\sin(\theta)\cos(\phi)-r\dot{\theta}^{2} \sin(\theta)\sin(\phi)\right.$    
  $\displaystyle \left. +2r\dot{\theta}\dot{\phi}\cos(\theta)\cos(\phi)+r\ddot{\th...
...ta)\sin(\phi)+r\ddot{\phi} \sin(\theta)\cos(\phi)\right] \cos(\theta)\sin(\phi)$    
  $\displaystyle -\left[ \ddot{r}\cos(\theta)-2\dot{r}\dot{\theta}\sin(\theta)-r\dot{\theta }\cos(\theta)-r\ddot{\theta}\sin(\theta)\right] \sin(\theta)$    
$\displaystyle =$ $\displaystyle \ddot{r}\left[ \sin(\theta)\cos(\phi)\cos(\theta)\cos(\phi)+\sin (\theta)\sin(\phi)\cos(\theta)\sin(\phi)-\cos(\theta)\sin(\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\theta}\left[ \cos(\theta)\cos(\phi)\cos(\theta)\co...
...)+\cos(\theta)\sin(\phi)\cos(\theta)\sin(\phi)+\sin(\theta)\sin (\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\phi}\left[ -\sin(\theta)\sin(\phi)\cos(\theta)\cos (\phi)+\sin(\theta)\cos(\phi)\cos(\theta)\sin(\phi)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\phi)\cos(\theta)\cos(...
...)-\sin(\theta)\sin(\phi)\cos(\theta)\sin(\phi)+\cos(\theta)\sin(\theta )\right]$    
  $\displaystyle +2r\dot{\phi}\dot{\theta}\left[ -\cos(\theta)\sin(\phi)\cos(\theta )\cos(\phi)+\cos(\theta)\cos(\phi)\cos(\theta)\sin(\phi)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ \cos(\theta)\cos(\phi)\cos(\theta)\cos(\phi )+\cos(\theta)\sin(\phi)\cos(\theta)\sin(\phi)+\sin(\theta)\sin(\theta )\right]$    
  $\displaystyle +r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\phi)\cos(\theta)\cos(\phi )-\sin(\theta)\sin(\phi)\cos(\theta)\sin(\phi)\right]$    
  $\displaystyle +r\ddot{\phi}\left[ -\sin(\theta)\sin(\phi)\cos(\theta)\cos(\phi )+\sin(\theta)\cos(\phi)\cos(\theta)\sin(\phi)\right]$    
$\displaystyle =$ $\displaystyle \ddot{r}\left[ \sin(\theta)\cos(\theta)\cos^{2}(\phi)+\sin(\theta )\cos(\theta)\sin^{2}(\phi)-\cos(\theta)\sin(\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\theta}\left[ \cos^{2}(\theta)\cos^{2}(\phi)+\cos^{2} (\theta)\sin^{2}(\phi)+\sin^{2}(\theta)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\theta)\cos^{2}(\phi )-\sin(\theta)\cos(\theta)\sin^{2}(\phi)+\cos(\theta)\sin(\theta)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ \cos^{2}(\theta)\cos^{2}(\phi)+\cos^{2}(\theta )\sin^{2}(\phi)+\sin^{2}(\theta)\right]$    
  $\displaystyle +r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\theta)\cos^{2}(\phi)-\sin (\theta)\cos(\theta)\sin^{2}(\phi)\right]$    
$\displaystyle =$ $\displaystyle \ddot{r}\left[ \sin(\theta)\cos(\theta)-\cos(\theta)\sin(\theta)\right]$    
  $\displaystyle +2\dot{r}\dot{\theta}\left[ \cos^{2}(\theta)+\sin^{2}(\theta)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\theta)+\cos(\theta)\sin (\theta)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ \cos^{2}(\theta)+\sin^{2}(\theta)\right]$    
  $\displaystyle -r\dot{\phi}^{2}\left[ \sin(\theta)\cos(\theta)\right]$    
$\displaystyle =$ $\displaystyle 2\dot{r}\dot{\theta}+r\ddot{\theta}-r\sin(\theta)\cos(\theta)\dot{\phi} ^{2}$    

und schliesslich

$\displaystyle a_{\phi} =$ $\displaystyle -\ddot{x}\sin(\phi)+\ddot{y}\cos(\phi)$ (G..586)
$\displaystyle =$ $\displaystyle -\left[ \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos (...
...dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta} ^{2}\sin(\theta)\cos(\phi)\right.$    
  $\displaystyle \left. -2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\th...
...{2}\sin(\theta)\cos(\phi)-r\ddot{\phi} \sin(\theta)\sin(\phi)\right] \sin(\phi)$    
  $\displaystyle +\left[ \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\...
...dot{\phi}\sin(\theta)\cos(\phi)-r\dot{\theta}^{2} \sin(\theta)\sin(\phi)\right.$    
  $\displaystyle \left. +2r\dot{\theta}\dot{\phi}\cos(\theta)\cos(\phi)+r\ddot{\th...
...{2}\sin(\theta)\sin(\phi)+r\ddot{\phi} \sin(\theta)\cos(\phi)\right] \cos(\phi)$    
$\displaystyle =$ $\displaystyle \ddot{r}\left[ -\sin(\theta)\cos(\phi)\sin(\phi)+\sin(\theta)\sin (\phi)\cos(\phi)\right]$    
  $\displaystyle +2\dot{r}\dot{\theta}\left[ -\cos(\theta)\cos(\phi)\sin(\phi)+\cos (\theta)\sin(\phi)\cos(\phi)\right]$    
  $\displaystyle +2\dot{r}\dot{\phi}\left[ \sin(\theta)\sin(\phi)\sin(\phi)+\sin(\theta )\cos(\phi)\cos(\phi)\right]$    
  $\displaystyle +r\dot{\theta}^{2}\left[ \sin(\theta)\cos(\phi)\sin(\phi)-\sin(\theta )\sin(\phi)\cos(\phi)\right]$    
  $\displaystyle +2r\dot{\phi}\dot{\theta}\left[ \cos(\theta)\sin(\phi)\sin(\phi )+\cos(\theta)\cos(\phi)\cos(\phi)\right]$    
  $\displaystyle +r\ddot{\theta}\left[ -\cos(\theta)\cos(\phi)\sin(\phi)+\cos(\theta )\sin(\phi)\cos(\phi)\right]$    
  $\displaystyle +r\dot{\phi}^{2}\left[ \sin(\theta)\cos(\phi)\sin(\phi)-\sin(\theta )\sin(\phi)\cos(\phi)\right]$    
  $\displaystyle +r\ddot{\phi}\left[ \sin(\theta)\sin(\phi)\sin(\phi)+\sin(\theta)\cos (\phi)\cos(\phi)\right]$    
$\displaystyle =$ $\displaystyle +2\dot{r}\dot{\phi}\left[ \sin(\theta)\sin^{2}(\phi)+\sin(\theta)\cos ^{2}(\phi)\right]$    
  $\displaystyle +2r\dot{\phi}\dot{\theta}\left[ \cos(\theta)\sin^{2}(\phi)+\cos(\theta )\cos^{2}(\phi)\right]$    
  $\displaystyle +r\ddot{\phi}\left[ \sin(\theta)\sin^{2}(\phi)+\sin(\theta)\cos^{2} (\phi)\right]$    
$\displaystyle =$ $\displaystyle +2\dot{r}\dot{\phi}\sin(\theta)+2r\dot{\phi}\dot{\theta}\cos(\theta )+r\ddot{\phi}\sin(\theta)$    
$\displaystyle =$ $\displaystyle \left[ r\ddot{\phi}+2\dot{r}\dot{\phi}\right] \sin(\theta)+2r\dot{\phi }\dot{\theta}\cos(\theta)$    

Zusammenfassend haben wir

$\displaystyle \vec{a}=$ $\displaystyle a_{r}\vec{e}_{r}+a_{\theta}\vec{e}_{\theta}+a_{\phi}\vec{e}_{\phi }$ (G..587)
$\displaystyle =$ $\displaystyle \left[ \ddot{r}-r\dot{\theta}^{2}-r\sin^{2}(\theta)\dot{\phi}^{2}\right] \vec{e}_{r}$    
  $\displaystyle +\left[ 2\dot{r}\dot{\theta}+r\ddot{\theta}-r\sin(\theta )\cos(\theta)\dot{\phi}^{2}\right] \vec{e}_{\theta}$    
  $\displaystyle +\left[ \left( r\ddot{\phi}+2\dot{r}\dot{\phi}\right) \sin(\theta)+2r\dot{\phi}\dot{\theta }\cos(\theta)\right] \vec{e}_{\phi}$    

Interpretation

Wir teilen die Beschleunigung in drei Komponenten auf

$\displaystyle \vec{a}= \vec{a}_p+\vec{a}_z+\vec{a}_c$ (G..588)

Dies ist in der angegebenen Reihenfolge die Parallelbeschleunigung, die den Betrag der Geschwindigkeit erhöht, die Zentripetalbeschleunigung und die Coriolis-Beschleunigung.

Im Einzelnen haben wir

$\displaystyle \vec{a}_p$ $\displaystyle = \ddot{r}\vec{e}_r+r\ddot{\theta}\vec{e}_\theta+r\sin(\theta)\ddot{\phi}\vec{e}_\phi$ (G..589)
$\displaystyle \vec{a}_z$ $\displaystyle = -r\left[\dot{\theta}^{2}+\sin^{2}(\theta)\dot{\phi}^{2}\right]\vec{e}_r -r\sin(\theta )\cos(\theta)\dot{\phi}^{2}\vec{e}_\theta$ (G..590)
$\displaystyle \vec{a}_c$ $\displaystyle = 2\dot{r}\dot{\theta}\vec{e}_\theta+2\left[\dot{r}\sin(\theta)+r\dot{\theta }\cos(\theta)\right]\dot{\phi}\vec{e}_\phi$ (G..591)

Othmar Marti
Experimentelle Physik
Universiät Ulm