$$ \begin{aligned} & \frac{d}{dt}\left( m_{2}l_{2}^{2}\dot{\theta}_{2}+4m_{2}l_{1}l_{2}\dot{\theta}_{1}\cos(\theta_{2}-\theta_{1}) \right)

\end{aligned} $$

根据链式法则，我们有：

$$ \begin{aligned} & \frac{d}{dt}\left( m_{2}l_{2}^{2}\dot{\theta}_{2}+4m_{2}l_{1}l_{2}\dot{\theta}_{1}\cos(\theta_{2}-\theta_{1}) \right) \\ =& m_{2}l_{2}^{2}\ddot{\theta}_{2}+4m_{2}l_{1}l_{2}\left(-\dot{\theta}_{1}\dot{\theta}_{2}\sin(\theta_{2}-\theta_{1})+\ddot{\theta}_{1}\cos(\theta_{2}-\theta_{1})-\dot{\theta}_{1}\dot{\theta}_{2}\sin(\theta_{2}-\theta_{1})-\dot{\theta}_{1}^{2}\sin(\theta_{2}-\theta_{1})\right) \\ =& m_{2}l_{2}^{2}\ddot{\theta}_{2}+4m_{2}l_{1}l_{2}\left(\ddot{\theta}_{1}\cos(\theta_{2}-\theta_{1})-\dot{\theta}_{1}^{2}\sin(\theta_{2}-\theta_{1})-\dot{\theta}_{1}\dot{\theta}_{2}\sin(\theta_{2}-\theta_{1})\right) \end{aligned} $$

其中，$m_2$ 是挂在杆上的质量，$l_1$ 和 $l_2$ 分别是两个杆的长度，$\theta_1$ 和 $\theta_2$ 分别是两个杆的角度，$\dot{\theta}_1$ 和 $\dot{\theta}_2$ 分别是两个杆的角速度，$\ddot{\theta}_1$ 和 $\ddot{\theta}_2$ 分别是两个杆的角加速度。