Problem $3.$ As shown in Figure $1,$ a planar double simple pendulum consists of particles $y_1$ and $y_2$ with masses $m_1$ and $m_2,$ respectively. The particle $y_1$ is connected to a frictionless pin joint at the point $w$ in the ceiling by means of the massless link $L_1$ with length $l_1,$ and the particle $y_2$ is connected by a frictionless pin joint at $y_1$ to the first link by means of the massless link $L_2$ with length $l_2.$ The external torque ${}_{\text{e}\text{x}\text{t}}$ is applied to link $L_1.$

Figure $1$: Planar double simple pendulum.

The equations of motion of the double pendulum are

$(m_2+m_1)l_1^2{}_1^{}+m_2{l}_1{l}_2(cos){}_2^{}=m_2{l}_1{l}_2(sin){}_2^{}{?}^2-(m_1+m_2)g{l}_{1}sin{}_1+{}_{\text{e}\text{x}\text{t}}$

$l_1(cos){}_1^{}+l_2{}_2^{}=-l_1(sin){}_1^{}{?}^2-$gsin${}_2$

where ${}_2-{}_1.$

Write $(13),(14)$ in the $x^{}=f(x)$ form.

Consider the case where ${}_1={}_2=0$ and ${}_1^{}={}_2^{}=0.$ Confirm that the double pendulum is at equilibrium in this configuration. Linearize the nonlinear differential equation about this equilibrium point to obtain the corresponding linearized system ${}^{}=A.$ Comment on the stability of the linearized system and the nonlinear system.

Consider the case where ${}_1={}_2=$ and ${}_1^{}={}_2^{}=0.$ Confirm that the double pendulum is at equilibrium in this configuration. Linearize the nonlinear differential equation about this equilibrium point to obtain the corresponding linearized system ${}^{}=A.$ Comment on the stability of the linearized system and the nonlinear system.

Using the Matlab routine ode$45,$ simulate the nonlinear system $x=f(x).$ Set $x(0)=x_{eq}+0\mathrm{.}1*$ randn $(4,1),$ where $x_{eq}$ is the equilibrium point in the previous parts and ${}_{\text{o}\text{x}\text{t}}=0.$ Simulate the system near both equilibrium points. Comment on the system's behavior and your prediction about the stability of the equilibrium point. Can you do it in Matlab?