Talk:PlanetPhysics/Polar Coordinate Motion Example of Generalized Coordinates

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: polar coordinate motion example of generalized coordinates %%% Primary Category Code: 45.20.Jj %%% Filename: PolarCoordinateMotionExampleOfGeneralizedCoordinates.tex %%% Version: 2 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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As an example let us get the equations in polar coordinates for \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} 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 \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} resolved along the \htmladdnormallink{radius vector}{http://planetphysics.us/encyclopedia/PositionVector.html}. $$ \frac{\partial T}{\partial\dot{\phi}}=m r^{2}\dot{\phi}, $$

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

$$ \delta_{\phi}W=m\frac{d}{dt}(r^{2}\dot{\phi})\delta\phi=\Phi r\delta\phi$$

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

In a more familiar form

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

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