polar coordinate motion example of generalized coordinates

As an example let us get the equations in polar coordinates for motion in a plane

Here

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

$\displaystyle \dot{x}^{2}+\dot{y}^{2}=v^{2}=\dot{r}^{2}+r^{2}\dot{\phi}^{2} $

and

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

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

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

$\displaystyle \delta_{r}W=m[\ddot{r}-r\dot{\phi}^{2}]\delta r=R\delta r
$

if $ R$ is the impressed force resolved along the radius vector.

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

$\displaystyle \frac{\partial T}{\partial \phi}=0.
$

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

if $ \Phi$ is the impressed force resolved perpendicular to the radius vector.

In a more familiar form

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

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



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