Chapter 5.6, Problem 8E

A double pendulum swinging in a vertical plane under the influence of gravity (see Figure 5.35) satisfies the system

$\begin{array}{rll}\left(m_1+m_2\right)l_1^2{\theta }_1{}^{\prime \text{}\prime }+m_2{l}_1{l}_2{\theta }_2{}^{\prime \text{}\prime }+\left(m_1+m_2\right)l_{1}g{\theta }_1& =& 0,\\ m_2{l}_2^2{\theta }_2{}^{\prime \text{}\prime }+m_2{l}_1{l}_2{\theta }_1{}^{\prime \text{}\prime }+m_2{l}_{2}g{\theta }_2& =& 0,\end{array}$

when ${\theta }_1$ and ${\theta }_2$ are small angles. Solve the system when $m_1=3\text{kg}$, $m_2=2\text{kg}$, $l_1=l_2=5\text{m}$, ${\theta }_1\left(0\right)=\pi /6$, ${\theta }_2\left(0\right)={\theta }_1{}^{\prime }\left(0\right)={\theta }_2\left(0\right)=0$.