SphericalCoordinateMotionExampleOfGeneralizedCoordinates 2008-07-18 20:53:42 2008-07-18 20:53:42 Example generalized coordinates for free motion % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here As an example let us get the equations in spherical coordinates for the motion. Where $$ x=r\cos\theta, \,\,\,\,\,\, y=r\sin\theta\cos\phi, \,\,\,\,\,\, z=r\sin\theta \sin \phi, $$ $$ T=\frac{m}{2}\left[\dot{r}^{2}+ r^2 \dot{\theta}^2 r^2 \sin^{2}\theta\dot{\phi}^{2}\right]. $$ $$ \frac{\partial T}{\partial\dot{r}}=m\dot{r}, $$ $$ \frac{\partial T}{\partial r}=m r\left[\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\right], $$ $$ \frac{\partial T}{\partial\dot{\theta}}=m r^{2}\dot{\theta}, $$ $$ \frac{\partial T}{\partial\theta}=m r^{2}\sin\theta\cos\theta\dot{\phi}^{2}, $$ $$ \frac{\partial T}{\partial\dot{\phi}}=m r^{2}\sin^{2}\theta\dot{\phi}. $$ $$ \delta_{r}W=m\left[\ddot{r}-r\left(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\right)\right] \delta r=R\delta r, $$ $$ \delta_{\theta}W=m\left[\frac{d}{dt}\left(r^{2}\dot{\theta}\right)-r^{2}\sin\theta\cos\theta\dot{\phi}^{2}\right] \delta\theta=\Theta r \delta \theta, $$ $$ \delta_{\phi}W=m\frac{d}{dt}\left(r^{2}\sin^{2}\theta\dot{\phi}\right)\delta\phi=\Phi r\sin\theta\delta\phi; $$ or $$m \left \{\frac{d^{2}r}{dt^{2}}-r\left[\left(\frac{d\theta}{dt}\right)^{2}+\sin^{2}\theta\left(\frac{d\phi}{dt}\right)^{2}\right]\right\}=R, $$ $$ \frac{m}{r}\left[\frac{d}{dt}\left(r^{2}\frac{d\theta}{dt}\right)-r^{2}\sin\theta\cos\theta\left(\frac{d\phi}{dt}\right)^2\right]=\Theta, $$ $$ \frac{m}{r\sin\theta}\frac{d}{dt}\left(r^{2}\sin^{2}\theta\frac{d\phi}{dt}\right)=\Phi. $$