双摆和三摆
双摆
（未完成）
三摆
Fig. 1：三摆
如fig. 1 所示，三根质量不计的杆长度分别为 $l_1, l_2, l_3$，三个质点的质量分别为 $m_1, m_2, m_3$．我们把这个模型叫做三摆．三摆常用于演示物理学中的混沌现象．
运动方程
\begin{equation}
\begin{aligned}
T &= \frac{1}{2} m_1 (l_1 \omega_1)^2 + \frac{1}{2} m_2 [(l_1 \omega_1 \cos\theta_1 + l_2 \omega_2 \cos\theta_2)^2 +\\
&\qquad \qquad\qquad\qquad\qquad\qquad (l_1 \omega_1 \sin\theta_1 + l_2 \omega_2 \sin\theta_2)^2]\\
& + \frac{1}{2} m_3 [(l_1 \omega_1 \cos\theta_1 + l_2 \omega_2 \cos\theta_2 + l_3 \omega_3 \cos \theta_3)^2 +\\
&\qquad\qquad\qquad (l_1 \omega_1 \sin\theta_1 + l_2 \omega_2 \sin\theta_2 + l_3 \omega_3 \sin\theta_3)^2]
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
V &= -m_1 g l_1 \cos \theta_1 - m_2 g (l_1\cos \theta_1 + l_2 \cos \theta_2)\\
&\qquad - m_3 g (l_1 \cos\theta_1 + l_2 \cos \theta_2 + l_3 \cos \theta_3)
\end{aligned}
\end{equation}
\begin{equation}
L(\theta_1, \theta_2, \theta_3, \omega_1, \omega_2, \omega_3) = T(\omega_1, \omega_2, \omega_3) - V(\theta_1, \theta_2, \theta_3)
\end{equation}
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{t}} \frac{\partial L}{\partial \omega_i} = \frac{\partial L}{\partial \theta_i} \quad (i=1,2,3)
\end{equation}
这是一个二阶微分方程组．