Talk:PlanetPhysics/Path Independence of Work
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[edit source]%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: path independence of work %%% Primary Category Code: 45.50.-j %%% Filename: PathIndependenceOfWork.tex %%% Version: 1 %%% Owner: mdo %%% Author(s): mdo %%% 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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Suppose an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/Mass.html} $m$ is free to move in some \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html}, $D$ (it is assumed that $D\subseteq\mathbb{R}^{3}$), and let $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ denote the \htmladdnormallink{position vectors}{http://planetphysics.us/encyclopedia/PositionVector.html} of points in $D$. The \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} required to move the object from $\mathbf{r}_{1}$ to $\mathbf{r}_{2}$ is given by
\begin{equation} W_{12} = \int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{F}\cdot d\mathbf{r}, \end{equation} where $\mathbf{F}$ is the total \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} acting on the object, as a \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} in $D$. If $\mathbf{F}$ is a conservative force, then it can be expressed in terms of a potential function; in particular, if $U$ is taken to denote the potential \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, then \begin{equation} \mathbf{F} = -\nabla U, \end{equation} where $\nabla$ denotes the \htmladdnormallink{gradient operator}{http://planetphysics.us/encyclopedia/Gradient.html}. Under such conditions, the work required to move the object of mass $m$ from position $\mathbf{r}_{1}$ to $\mathbf{r}_{2}$ in $D$ is path independent. This means that if the object were to move along a straight line connecting $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$, the amount of work done would be in exact equality with any other path.
\subsection{Proof of Path Independence} Given the expression for work, \begin{equation} W_{12} = \int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{F}\cdot d\mathbf{r}, \end{equation} and the \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} between the conservative force, $\mathbf{F}$ and the potential energy, $U$, \begin{equation} \mathbf{F} = -\nabla U, \end{equation} it follows that, upon substitution of the later into the former, \begin{eqnarray*} W_{12} & = & \int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{F}\cdot d\mathbf{r}\\ & = & -\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\nabla U\cdot d\mathbf{r}. \end{eqnarray*} Focus on the integrand, $\nabla U\cdot d\mathbf{r}$, and write it in terms of its components as, \begin{eqnarray*} \nabla U\cdot d\mathbf{r} & = & \left( \frac{\partial U}{\partial x_{1}}, \frac{\partial U}{\partial x_{2}}, \frac{\partial U}{\partial x_{3}}\right) \cdot\left( dx_{1}, dx_{2}, dx_{3}\right)\\ & = & \frac{\partial U}{\partial x_{1}}dx_{1} + \frac{\partial U}{\partial x_{2}}dx_{2} + \frac{\partial U}{\partial x_{3}}dx_{3}. \end{eqnarray*} Now, recall that for some arbitrary function, $f=f(x,y,z)$, the differential of that function is $$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz.$$ Based on this, it immediately follows that \begin{equation} \nabla U\cdot d\mathbf{r} = dU. \end{equation} Substituting this result back into the work equation, \begin{eqnarray} W_{12} & = & \int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{F}\cdot d\mathbf{r}\\ & = & \int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}dU\\ & = & U(\mathbf{r}_{2}) - U(\mathbf{r}_{1}). \end{eqnarray} Therefore, from the final equation, it is clearly seen that the work to move the object from position $\mathbf{r}_{1}$ to $\mathbf{r}_{2}$ is only dependent upon the potential energy at those positions, and not the path taken. Note that in the above, the minus sign in front of the integral has been dropped; this was done to show, in the final result, the amount of work done by the \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}. That is, if the potential energy at the final position is greater than that at the initial, then $W_{12}$ is positive, and has \emph{done} work.
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