spherical coordinate motion example of generalized coordinates

As an example let us get the equations in spherical coordinates for the motion.

Where

$\displaystyle x=r\cos\theta, \,\,\,\,\,\, y=r\sin\theta\cos\phi, \,\,\,\,\,\, z=r\sin\theta \sin \phi,
$

$\displaystyle T=\frac{m}{2}\left[\dot{r}^{2}+ r^2 \dot{\theta}^2 r^2 \sin^{2}\theta\dot{\phi}^{2}\right].
$

$\displaystyle \frac{\partial T}{\partial\dot{r}}=m\dot{r},
$

$\displaystyle \frac{\partial T}{\partial r}=m r\left[\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\right],
$

$\displaystyle \frac{\partial T}{\partial\dot{\theta}}=m r^{2}\dot{\theta},
$

$\displaystyle \frac{\partial T}{\partial\theta}=m r^{2}\sin\theta\cos\theta\dot{\phi}^{2},
$

$\displaystyle \frac{\partial T}{\partial\dot{\phi}}=m r^{2}\sin^{2}\theta\dot{\phi}.
$

$\displaystyle \delta_{r}W=m\left[\ddot{r}-r\left(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\right)\right] \delta r=R\delta r,
$

$\displaystyle \delta_{\theta}W=m\left[\frac{d}{dt}\left(r^{2}\dot{\theta}\right...
...\sin\theta\cos\theta\dot{\phi}^{2}\right] \delta\theta=\Theta r \delta \theta,
$

$\displaystyle \delta_{\phi}W=m\frac{d}{dt}\left(r^{2}\sin^{2}\theta\dot{\phi}\right)\delta\phi=\Phi r\sin\theta\delta\phi;
$

or

$\displaystyle m \left \{\frac{d^{2}r}{dt^{2}}-r\left[\left(\frac{d\theta}{dt}\right)^{2}+\sin^{2}\theta\left(\frac{d\phi}{dt}\right)^{2}\right]\right\}=R,
$

$\displaystyle \frac{m}{r}\left[\frac{d}{dt}\left(r^{2}\frac{d\theta}{dt}\right)-r^{2}\sin\theta\cos\theta\left(\frac{d\phi}{dt}\right)^2\right]=\Theta,
$

$\displaystyle \frac{m}{r\sin\theta}\frac{d}{dt}\left(r^{2}\sin^{2}\theta\frac{d\phi}{dt}\right)=\Phi.
$



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