PlanetPhysics/Polar Coordinate Motion Example of Generalized Coordinates

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

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

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

and

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

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

$$ \frac{\partial T}{\partial r}=m r \dot{\phi}^{2}. $$ $$ \delta_{r}W=m[\ddot{r}-r\dot{\phi}^{2}]\delta r=R\delta r $$ if $$R$$ is the impressed force resolved along the radius vector. $$ \frac{\partial T}{\partial\dot{\phi}}=m r^{2}\dot{\phi}, $$

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

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

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

In a more familiar form

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