| 1656 |
|
\end{array} |
| 1657 |
|
\] |
| 1658 |
|
, we obtain |
| 1659 |
< |
\begin{eqnarray*} |
| 1660 |
< |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
| 1659 |
> |
\[ |
| 1660 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1661 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1662 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
| 1663 |
< |
_\alpha t)\dot x(t - \tau )d\tau \\ |
| 1664 |
< |
& & - \left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
| 1665 |
< |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) - |
| 1666 |
< |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 1667 |
< |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} \\ |
| 1668 |
< |
% |
| 1669 |
< |
& = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1663 |
> |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
| 1664 |
> |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
| 1665 |
> |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 1666 |
> |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
| 1667 |
> |
\] |
| 1668 |
> |
\[ |
| 1669 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1670 |
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1671 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 1672 |
|
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
| 1673 |
< |
{\left[ {g_\alpha x_\alpha (0) \\ |
| 1674 |
< |
& & - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
| 1675 |
< |
(\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 1676 |
< |
_\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1677 |
< |
\end{eqnarray*} |
| 1673 |
> |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
| 1674 |
> |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
| 1675 |
> |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
| 1676 |
> |
(\omega _\alpha t)} \right\}} |
| 1677 |
> |
\] |
| 1678 |
> |
|
| 1679 |
|
Introducing a \emph{dynamic friction kernel} |
| 1680 |
|
\begin{equation} |
| 1681 |
|
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
