--- trunk/tengDissertation/Introduction.tex	2006/06/05 21:24:52	2793
+++ trunk/tengDissertation/Introduction.tex	2006/06/06 01:58:27	2794
@@ -1656,26 +1656,26 @@ transformations:
  \end{array}
 \]
 , we obtain
-\[
-m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
+\begin{eqnarray*}
+m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
 \sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
 }}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
-_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
-- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
-(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
-_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
-\]
-\[
-m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
+_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
+x_\alpha (0) - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}}
+\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
+\end{eqnarray*}
+\begin{eqnarray*}
+m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
 {\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
 }}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
-t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
-{\left[ {g_\alpha  x_\alpha  (0) - \frac{{g_\alpha  }}{{m_\alpha
-\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
-\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
-(\omega _\alpha  t)} \right\}}
-\]
-
+t)\dot x(t - \tau )d} \tau }  \\
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
+x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha  }}}
+\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
+\end{eqnarray*}
 Introducing a \emph{dynamic friction kernel}
 \begin{equation}
 \xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2