CylindricalCoordinateMotionExampleOfGeneralizedCoordinates 2008-07-18 20:25:21 2008-07-18 20:25:21 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 cylindrical coordinates $$ x=r\cos\phi, \,\,\,\,\,\, y=r\sin\phi, \,\,\,\,\,\, z=z, $$ $$ T=\frac{m}{2} \left[\dot{r}^{2}+r^2\dot{\phi}^{2}+\dot{z}^{2} \right]. $$ $$ \frac{\partial T}{\dot{r}}=m\dot{r}, $$ $$ \frac{T}{\partial r}=m r\dot{\phi}^{2}, $$ $$ \frac{\partial T}{\partial\dot{\phi}}=mr^{2}\dot{\phi}, $$ $$ \frac{\partial T}{\partial \dot{z}}=m\dot{z}. $$ $$ \delta_{r}W=m \left[\ddot{r} - r\dot{\phi}^{2} \right] \delta r=R\delta r, $$ $$ \delta_{\phi}W=m\frac{d}{dt} \left(r^{2}\dot{\phi}\right)\delta\phi=\Phi r\delta\phi, $$ $$ \delta_z W= m \ddot{z} \delta z = Z \delta z; $$ or $$m \left[ \frac{d^{2}r}{dt^{2}}-r \left(\frac{d\phi}{dt}\right)^{2}\right]=R,$$ $$ \frac{m}{r}\frac{d}{dt}\left(r^{2}\frac{d\phi}{dt}\right)=\Phi, $$ $$ m\frac{d^{2}z}{dt^{2}}=Z. $$