PlanetPhysics/Spherical Coordinate Motion Example of Generalized Coordinates

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

Where

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

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

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

$\frac{\partial T}{\partial r}=mr[{\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2],$

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

$\frac{\partial T}{\partial \theta }=m{r}^2\sin \theta \cos \theta {\dot{\varphi }}^2,$

$\frac{\partial T}{\partial \dot{\varphi }}=m{r}^2{\sin }^2\theta \dot{\varphi }.$

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

${\delta }_{\theta }W=m[\frac{d}{dt}\left(r^2\dot{\theta }\right)-r^2\sin \theta \cos \theta {\dot{\varphi }}^2]\delta \theta =\mathrm{\Theta }r\delta \theta ,$

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

or

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

$\frac{m}{r}[\frac{d}{dt}\left(r^2\frac{d\theta }{dt}\right)-r^2\sin \theta \cos \theta {\left(\frac{d\varphi }{dt}\right)}^2]=\mathrm{\Theta },$

$\frac{m}{r\sin \theta }\frac{d}{dt}\left(r^2{\sin }^2\theta \frac{d\varphi }{dt}\right)=\mathrm{\Phi }.$