The problem presents a comprehensive analysis of the dynamics and stability of a planar double simple pendulum system. The system comprises two particles, $\backslash ($y$\_1\backslash )$ and $\backslash ($y$\_2\backslash ),$ with masses $\backslash ($m$\_1\backslash )$ and $\backslash ($m$\_2\backslash ),$ respectively, connected by massless links $\backslash (\backslash$mathcal$\{$L$\}\_\{1\}\backslash )$ and $\backslash (\backslash$mathcal$\{$L$\}\_\{2\}\backslash )$ of lengths $\backslash (\backslash$ell$\_\{1\}\backslash )$ and $\backslash (\backslash$ell$\_\{2\}\backslash ),$ respectively. The first particle is attached to a ceiling point $\backslash ($w$\backslash )$ via a frictionless pin joint, and the second particle is similarly connected to the first. An external torque $\backslash (\backslash$tau$\_\{\backslash$text$\{$ext$\}\}\backslash )$ is applied to $\backslash (\backslash$mathcal$\{$L$\}\_\{1\}\backslash ).$ The equations of motion for this system are derived, incorporating gravitational forces and the torque, and are expressed in terms of the angular positions $\backslash (\backslash$theta$\_1\backslash )$ and $\backslash (\backslash$theta$\_2\backslash ),$ and their second derivatives $($accelerations$)\mathrm{.}1.**$Conversion to $\backslash (\backslash$dot$\{$x$\}=$f$($x$)\backslash )$ Form$**$: The equations of motion are initially presented in a form that highlights the system's dynamics in terms of angular accelerations. To convert these into the standard $\backslash (\backslash$dot$\{$x$\}=$f$($x$)\backslash )$ form, one needs to express the system's state derivatives $($i$.$e$.,$ the velocities and accelerations of $\backslash (\backslash$theta$\_1\backslash )$ and $\backslash (\backslash$theta$\_2\backslash ))$ as functions of the system's current state $($i$.$e$.,$ the angles and their velocities$).$ This involves rearranging the given equations to isolate the derivatives of the state variables on one side, effectively transforming the system into a set of first$-$order differential equations that describe the time evolution of the system's state$\mathrm{.}2.**$Equilibrium Analysis and Linearization$**$: $-$ At $\backslash (\backslash$theta$\_\{1\}=\backslash$theta$\_\{2\}=0\backslash ),$ the system is confirmed to be in equilibrium by showing that, in this configuration, the accelerations $\backslash (\backslash$ddot$\{\backslash$theta$\}\_1\backslash )$ and $\backslash (\backslash$ddot$\{\backslash$theta$\}\_2\backslash )$ are zero, satisfying the equilibrium condition where the net forces $($and hence accelerations$)$ acting on the system are zero. Linearizing the system about this equilibrium involves approximating the nonlinear dynamics by a linear system around the equilibrium point, resulting in a linearized system represented by $\backslash (\backslash$dot$\{\backslash$xi$\}=$A $\backslash$xi $\backslash ).$ The stability of this linearized system, and by extension insights into the stability of the original nonlinear system, can be assessed by examining the eigenvalues of the matrix $\backslash ($A$\backslash ).-$ The case where $\backslash (\backslash$theta$\_\{1\}=\backslash$theta$\_\{2\}=\backslash$pi $\backslash )$ represents another equilibrium configuration. The linearization process is similar, aiming to understand the system's behavior in the vicinity of this equilibrium point. The stability analysis again involves evaluating the eigenvalues of the linearized system's matrix$\mathrm{.}3.**$Simulation and Stability Prediction$**$: $-$ The nonlinear system $\backslash (\backslash$dot$\{$x$\}=$f$($x$)\backslash )$ is simulated using Matlab's $`$ode$45`$ routine, a numerical solver for ordinary differential equations. The initial conditions for the simulation are set to a small perturbation around the equilibrium points discussed, with $\backslash (\backslash$tau$\_\{\backslash$text$\{$ext$\}\}=0\backslash ).$ This simulation aims to observe the system's response to small disturbances from its equilibrium states, thereby providing insights into the system's stability. $-$ The behavior of the system near both equilibrium points is analyzed through these simulations. The stability of each equilibrium point is predicted based on how the system responds to perturbationswhether it returns to equilibrium $($indicating stability$)$ or diverges further away $($indicating instability$).$In summary, the problem involves a detailed examination of the dynamics of a double pendulum system, including deriving its equations of motion, transforming these into a form suitable for analysis and simulation, assessing the system's equilibrium configurations and their stability, and using numerical simulation to observe the system's behavior in response to perturbations. The analysis combines theoretical dynamics with computational tools to provide a comprehensive understanding of the double pendulum's behavior under various conditions. Can you do it in Matlab?