TransformationFromRectangularToGeneralizedCoordinates 2008-07-15 10:48:33 2008-07-15 10:48:33 Topic % 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 We take a system 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 generalized coordinates we denote by $q_\alpha$ $(l \le \alpha \le \nu )$. Since the generalized coordinates completely specify the position of their system, $x_i$ are their unique functions: $$x_i = x_i (q_1, q_2,\dots q_\alpha,\dots,q_v)$$ From this it is easy to obtain an expression for the Cartesian components of velocity. 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 Einstein 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 acceleration: $$\ddot{x_i}= \frac{d}{dt}\left( \frac{\partial x_i}{\partial q_\alpha} \right ) \dot{q_\alpha} + \frac{\partial x_i}{\partial q_\alpha} \ddot{q_\alpha} $$ 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 summatino 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 $\beta$.