Talk:PlanetPhysics/Transformation From Rectangular to Generalized Coordinates

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: transformation from rectangular to generalized coordinates %%% Primary Category Code: 45.20.Jj %%% Filename: TransformationFromRectangularToGeneralizedCoordinates.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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We take a \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} with a total of $3N \equiv n$ Cartesian coordinates of which $\nu$ are independent. We denote Cartesian coordinates by the same letter $x_i$, understanding by this symbol all the coordinates $x, y, z$; this means that $i$ varies from $1$ to $3N$, that is, from $1$ to $n$. The \htmladdnormallink{generalized coordinates}{http://planetphysics.us/encyclopedia/CommutativeRingWithUnit.html} we denote by $q_\alpha$ $(l \le \alpha \le \nu )$. Since the generalized coordinates completely specify the \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} of their system, $x_i$ are their unique \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html}:

$$x_i = x_i (q_1, q_2,\dots q_\alpha,\dots,q_v)$$

From this it is easy to obtain an expression for the Cartesian components of \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html}. Differentiating the function of many variables $x_i(\dots q_\alpha)$ with respect to time, we have

\begin{equation} \frac{ dx_i}{dt} = \sum_{\alpha=1}^{\nu} \frac{\partial x_i}{\partial q_\alpha} \frac{d q_\alpha}{dt} \end{equation}

In the subsequent derivation we shall often have to perform summations with respect to all the generalized coordinates $q_\alpha$, and double and triple sums will be encountered. In order to save space we will use \htmladdnormallink{Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} summation.

The total derivative with respect to time is usually denoted by a dot over the corresponding variable:

$$ \frac{d x_i}{dt} = \dot{x_i}; \,\,\, \frac{d q_\alpha}{dt} = \dot{q_\alpha} $$

In this notation, the velocity (1) in abbreviated form becomes:

\begin{equation} \dot{x_i} = \frac{\partial x_i}{\partial q_\alpha} \dot{q_\alpha} \end{equation}

Differentiating this with respect to time again, we obtain an expression for the Cartesian components of \htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html}:

$$\ddot{x_i}= \frac{d}{dt}\left( \frac{\partial x_i}{\partial q_\alpha} \right ) \dot{q_\alpha} + \frac{\partial x_i}{\partial q_\alpha} \ddot{q_\alpha} $$

The total derivative in the first term is written as usual:

$$ \frac{d}{dt}\left( \frac{\partial x_i}{\partial q_\alpha} \right ) = \frac{\partial^2 x_i}{\partial q_\beta \partial q_\alpha} \dot{q_\beta} $$

The Greek symbol over which the summation is performed is deonted by the letter $\beta$ to avoid confusion with the symbol $\alpha$, which denotes the summation in the expression for velocity (2). Thus we obtain the desired expression for $\ddot{x_i}$:

\begin{equation} \ddot{x_i} = \frac{\partial^2 x_i}{\partial q_\beta \partial q_\alpha} \dot{q_\beta} \dot{q_\alpha} + \frac{\partial x_i}{\partial q_\alpha} \ddot{q_\alpha} \end{equation}

The first term on the right-hand side contains a double summation with respect to $\alpha$ and $\beta$.

\subsection{References}

[1] Kompaneyets, A. "\htmladdnormallink{Theoretical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html}." Foreign Languages Publishing House, Moscow, 1961.

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} [1]

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