PlanetPhysics/Cylindrical Coordinate Motion Example of Generalized Coordinates

As an example let us get the equations in cylindrical coordinates

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

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

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

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

$$ \frac{\partial T}{\partial\dot{\phi}}=mr^{2}\dot{\phi}, $$ $$ \frac{\partial T}{\partial \dot{z}}=m\dot{z}. $$

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

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

$$ \delta_z W= m \ddot{z} \delta z = Z \delta z; $$

or

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

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