PolarCoordinateMotionExampleOfGeneralizedCoordinates 2008-07-18 20:11:07 2008-07-18 20:11:07 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 polar coordinates for motion in a plane Here $$ x=r\cos\phi,\,\,\,\,\,\,\,\,\, y=r\sin\phi$$ $$ \dot{x}^{2}+\dot{y}^{2}=v^{2}=\dot{r}^{2}+r^{2}\dot{\phi}^{2} $$ and $$ T = \frac{m}{2}\left [ \dot{r}^{2}+r^{2}\dot{\phi}^{2} \right ] $$ $$ \frac{\partial T}{\partial \dot{r} } = m \dot{r} $$ $$ \frac{\partial T}{\partial r}=m r \dot{\phi}^{2}. $$ $$ \delta_{r}W=m[\ddot{r}-r\dot{\phi}^{2}]\delta r=R\delta r $$ if $R$ is the impressed force resolved along the radius vector. $$ \frac{\partial T}{\partial\dot{\phi}}=m r^{2}\dot{\phi}, $$ $$ \frac{\partial T}{\partial \phi}=0. $$ $$ \delta_{\phi}W=m\frac{d}{dt}(r^{2}\dot{\phi})\delta\phi=\Phi r\delta\phi$$ if $\Phi$ is the impressed force resolved perpendicular to the radius vector. In a more familiar form $$ m \left [\frac{d^{2}r}{dt^{2}}-r \left ( \frac{d\phi}{dt} \right)^2 \right ]=R, $$ $$ \frac{m}{r}\frac{d}{dt} \left ( r^{2}\frac{d\phi}{dt} \right )=\Phi. $$