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\documentclass[12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{graphicx}
\title{Equations For Paper}
\author{Cindy Harnett}
\begin{document}
\maketitle
\begin{abstract}
geeblegobble geeblegobble
\end{abstract}
\pagenumbering{gobble}
\section{Introduction}
mr smoddlebogy says work toward this equation by explaining and defining arclength, dx/dtheta etc
\section{Some Stuff}
\label{sec:}
\subsection{Relationship of moment and curvature}
Distinguish small-deflection and large-deflection solutions here
\subsection {Eq. 1: Euler-Bernoulli moment-curvature relationship}
\newpage
$$M=EI\frac{d\theta}{ds}$$
\newpage
\subsection{Eq. 2: Differential Arclength in terms of angle}
\newpage
$$\frac{ds}{d\theta}=\pm \sqrt{\frac{EI}{R}} \frac{1}{\sqrt{cos(\theta-\phi)-cos(\theta^{\ast}-\phi)}} $$
\newpage
\subsection{Eq. 5}
\newpage
$$s(\theta)=\sqrt{\frac{EI}{R}} \int_{\alpha}^{\theta}\frac{d\theta'}{\sqrt{cos(\theta'-\phi)-cos(\theta^{\ast}-\phi)}}$$
\newpage
\subsection{Eq. 6}
\newpage
$$x(\theta)=\sqrt{\frac{EI}{R}} \int_{\alpha}^{\theta} \frac{cos(\theta')d\theta'}{\sqrt{cos(\theta'-\phi)-cos(\theta^{\ast}-\phi)}}$$
\newpage
\subsection{Eq. 7}
\newpage
$$y(\theta)=\sqrt{\frac{EI}{R}} \int_{\alpha}^{\theta} \frac{sin(\theta')d\theta'}{\sqrt{cos(\theta'-\phi)-cos(\theta^{\ast}-\phi)}}$$
\newpage
\subsection{Eq. 8}
\newpage
$$\frac{L}{w}=\beta=\frac{\int_{\alpha}^{\theta^{\ast}}\frac{d\theta}{\sqrt {\cos \left( \theta -\phi \right) -\cos \left( \theta^{\ast } -\phi \right) }}-\int_{\theta^{\ast}}^{0}\frac{d\theta}{\sqrt {\cos \left( \theta -\phi \right) -\cos \left( \theta ^{\ast } -\phi\right) }}}{\int_{\alpha}^{\theta^{\ast}}\frac{cos(\theta) d\theta}{\sqrt {\cos \left( \theta -\phi \right) -\cos \left( \theta^{\ast } -\phi \right) }}-\int_{\theta^{\ast}}^{0}\frac{cos(\theta) d\theta}{\sqrt {\cos \left( \theta -\phi \right) -\cos \left( \theta^{\ast } -\phi \right) }}}$$
\newpage
\subsection{Eq. 9 Thing I integrated to get critical alpha}
\newpage
$$\beta=\frac{\int_{\alpha}^{0}\frac{d\theta}{\sqrt {\cos \left( \theta \right) -\cos \left( \alpha \right) }}}{\int_{\alpha}^{0}\frac{cos(\theta) d\theta}{\sqrt {\cos \left( \theta \right) -\cos \left( \alpha \right) }}} $$
\newpage
\subsection{Eq. 10 Bending energy in terms of M, it's in the word doc but could look nicer}
\newpage
$$du=\left( \frac{M^2}{2EI}\right) ds$$
\newpage
\subsection {Eq. 11 Have in another document: bending energy integral, already pasted in}
\newpage
$$U=\frac{EI}{2}\int \left( \frac{d\theta}{ds} \right) ^2\,ds $$
\newpage
\subsection {Eq. 12 Energy in terms of dtheta --this should contain R}
\newpage
$$U = \frac{\sqrt{REI}}{2}\int \sqrt{cos(\theta-\phi)-cos(\theta^{\ast}-\phi)}\,d\theta$$
\newpage
\subsection {Eq. 13 Expression for R}
\newpage
$$\sqrt{R}=\sqrt{EI} \int_{\alpha}^{0} \frac{cos(\theta)d\theta}{\sqrt{cos(\theta-\phi)-cos(\theta^{\ast}-\phi)}}$$
\newpage
\subsection{Lists}
You can make lists with automatic numbering \dots
\begin{enumerate}
\item Like this,
\item and like this.
\end{enumerate}
\dots or bullet points \dots
\begin{itemize}
\item Like this,
\item and like this.
\end{itemize}
\end{document}