Talk:PlanetPhysics/Generalized Coordinates for Free Motion

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: generalized coordinates for free motion %%% Primary Category Code: 45.20.Jj %%% Filename: GeneralizedCoordinatesForFreeMotion.tex %%% Version: 6 %%% 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}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

% 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

\begin{document}

The \htmladdnormallink{differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} for the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of a \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} under any \htmladdnormallink{forces}{http://planetphysics.us/encyclopedia/Thrust.html} when we use rectangular coordinates are known from Newston's laws of motion

$$ m \ddot{x} = F_x $$ $$ m \ddot{y} = F_y $$ $$ m \ddot{z} = F_z $$

where $F_x, F_y, F_z$ are the components of the actual forces on the particle resolved parallel to each of the fixed rectangular axes, or rather their equivalents $m \ddot{x}, m \ddot{y}, m \ddot{z}$, are called the \emph{effective forces} on the particle. They are of course a set of forces mechanically equivalent to the actual forces acting on the particle.

The equations of motion of the particle in terms of any other \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of coordinates are easily obtained.

Let $q_1, q_2, q_3$, be the coordinates in question. The appropriate \htmladdnormallink{formulas}{http://planetphysics.us/encyclopedia/Formula.html} for transformation of coordinates express $x,y,z$ in terms of $q_1,q_2,q_3$.

$$ x = f_1(q_1,q_2,q_3), \,\,\,\, y = f_2(q_1,q_2,q_3), \,\,\,\, z = f_3(q_1,q_2,q_3) $$

For the component \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} $\dot{x}$ we have

$$ \dot{x} = \frac{\partial x}{\partial q_1} \dot{q_1} + \frac{\partial x}{\partial q_2} \dot{q_2} + \frac{\partial x}{\partial q_3} \dot{q_3} $$

and $\dot{x},\dot{y},\dot{z}$ are explicit \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} of $q_1,q_2,q_3,\dot{q_1},\dot{q_2},\dot{q_3}$ linear and homogeneous in terms of $\dot{q_1},\dot{q_2},\dot{q_3}$.

We may note in passing that it follows from this fact that $\dot{x}^2,\dot{y}^2,\dot{z}^2$ are homogeneous quadratic functions of $\dot{q_1},\dot{q_2},\dot{q_3}$.

Obviously

$$ \frac{\partial \dot{x}}{\partial \dot{q_1}} = \frac{\partial x}{\partial q_1} $$

and since

$$ \frac{d}{dt} \frac{\partial x}{\partial q_1} = \frac{\partial^2 x}{\partial q_1^2} \dot{q_1} + \frac{\partial^2 x}{\partial q_2 \partial q_1} \dot{q_2} + \frac{\partial^2 x}{\partial q_3 \partial q_1} \dot{q_3} $$

and

$$ \frac{\partial \dot{x}}{\partial q_1} = \frac{\partial^2 x}{\partial q_1^2} \dot{q_1} + \frac{\partial^2 x}{\partial q_1 \partial q_2} \dot{q_2} + \frac{\partial^2 x}{\partial q_1 \partial q_3} \dot{q_3} $$

$$ \frac{d}{dt} \frac{\partial x}{\partial q_1} = \frac{\partial \dot{x}}{\partial q_1} $$

Let us now find an expression for the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} $\delta_{q_1}W$ done by the effective forces when the coordinate $q_1$ is changed by an infinitesimal amount $\delta q_1$ without changing $q_2$ or $q_3$. If $\delta x, \delta y, \delta z$ are changes thus produced in $x,y,z$, obviously from the definition of work

$$ \delta_{q_1}W = m\left [ \ddot{x} \delta x + \ddot{y} \delta y + \ddot{z} \delta z \right ]$$

if expressed in rectangular coordinates. We need, however, to express $\delta_{q_1}W$ in terms of our coordinates $q_1, q_2, q_3$.

$$ \delta_{q_1}W = m\left [ \ddot{x} \frac{\partial x}{\partial q_1} + \ddot{y} \frac{\partial y}{\partial q_1} + \ddot{z} \frac{\partial z}{\partial q_1} \right ]\delta q_1$$

Now

$$\ddot{x} \frac{\partial x}{\partial q_1} = \frac{d}{dt} \left ( \dot{x} \frac{\partial x}{\partial q_1} \right ) - \dot{x} \frac{d}{dt} \frac{\partial x}{\partial q_1} $$

but from earlier definitions

$$ \frac{\partial x}{\partial q_1} = \frac{\partial \dot{x}}{\partial \dot{q_1}} \,\,\,\,\, and \,\,\,\,\, \frac{d}{dt} \frac{\partial x}{\partial q_1} = \frac{\partial \dot{x}}{\partial q_1} $$

Hence

$$ \ddot{x} \frac{\partial x}{\partial q_1} = \frac{d}{dt} \left ( \dot{x} \frac{\partial \dot{x}}{\partial \dot{q_1}} \right ) - \dot{x} \frac{\partial \dot{x}}{\partial q_1} = \frac{d}{dt} \frac{\partial}{\partial \dot{q_1}} \left ( \frac{\dot{x}^2}{2} \right) - \frac{\partial}{\partial q_1} \left ( \frac{\dot{x}^2}{2} \right ) $$

and therefore

\begin{equation} \delta_{q_1} W = \left [ \frac{d}{dt} \frac{\partial T}{\partial \dot{q_1}} - \frac{\partial T}{\partial q_1} \right ] \delta q_1 \end{equation}

where

$$ T = \frac{m}{2} \left [ \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right ] $$

and is the \htmladdnormallink{kinetic energy}{http://planetphysics.us/encyclopedia/KineticEnergy.html} of the particle.

To get our differential equation we have only to write the second member of (1) equal to the work done by the actual forces when $q_1$ is changed by $\delta q_1$.

If we represent the work in question by $Q_1 \delta q_1$, our equation is

\begin{equation} \frac{d}{dt} \frac{\partial T}{\partial \dot{q_1}} - \frac{\partial T}{\partial q_1} = Q_1 \end{equation}

and of course we get such an equation for every coordinate. Even though we derived this differential equation for a single particle in free motion, it is the same for a systems of particles, except the kinetic energy is for all the particles in the system, which brings us to \htmladdnormallink{Lagrange's equations}{http://planetphysics.us/encyclopedia/Lagrangian.html} \begin{equation} Q_i = \frac{d}{dt} \left ( \frac{ \partial T}{\partial \dot{q_i}} \right ) - \frac{\partial T}{\partial q_i} \end{equation}

In any concrete problem, $T$ must be expressed in terms of $q_1,q_2,q_3$, and their time derivatives before we can form the expression for the work done by the effective forces. $Q_1 \delta q_1, Q_2 \delta q_2, Q_3 \delta q_3$, the work done by the actual forces, must be obtained from direct examination of the problem.

\end{document}