PlanetPhysics/Polar Coordinate Motion Example of Generalized Coordinates

As an example let us get the equations in polar coordinates for [[../CosmologicalConstant/|motion]] in a plane

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

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

and

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

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

$\frac{\partial T}{\partial r}=mr{\dot{\varphi }}^2.$ ${\delta }_{r}W=m[\ddot{r}-r{\dot{\varphi }}^2]\delta r=R\delta r$ if $R$ is the impressed [[../Thrust/|force]] resolved along the [[../PositionVector/|radius vector]]. $\frac{\partial T}{\partial \dot{\varphi }}=m{r}^2\dot{\varphi },$

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

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

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

In a more familiar form

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