PlanetPhysics/Cylindrical Coordinate Motion Example of Generalized Coordinates

As an example let us get the equations in cylindrical coordinates

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

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

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

$\frac{T}{\partial r}=mr{\dot{\varphi }}^2,$

$\frac{\partial T}{\partial \dot{\varphi }}=m{r}^2\dot{\varphi },$ $\frac{\partial T}{\partial \dot{z}}=m\dot{z}.$

${\delta }_{r}W=m[\ddot{r}-r{\dot{\varphi }}^2]\delta r=R\delta r,$

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

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

or

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

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