Talk:PlanetPhysics/Generalized Coordinates for Constrained Motion

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: generalized coordinates for constrained motion %%% Primary Category Code: 45.20.Jj %%% Filename: GeneralizedCoordinatesForConstrainedMotion.tex %%% Version: 1 %%% 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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If the \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} is constrained to move on some given surface, any two independent specified \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} of its rectangular coordinates $x, y, z$, may be taken as its coordinates $q_1$ and $q_2$, provided that by the equation of the given surface in rectangular coordinates and the equations formed by writing $q_1$ and $q_2$ equal to their values in terms of $x, y$, and $z$ the last-named coordinates may be uniquely obtained as explicit functions of $q_{1}$ and $q_2$.

If the particle is constrained to move in a given path, any specified function of $x,y,z$ may be taken as its coordinate $q_1$, provided that by the two rectangular equations of its path and the equation formed by writing $q_1$ equal to its value in terms of $x,y,z$ the last-named coordinates may be uniquely obtained as explicit functions of $q_1$.

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