$5.$ Use Lagrangian mechanics to numerically simulate the concoction below. It consists of a rigid fixed halfpipe of radius $\backslash ($ R $\backslash )$ with two solid cylinders inside it having radius $\backslash ($ r $\backslash )$ and mass $\backslash ($ m $\backslash ).$ The surface of the half$-$pipe is sufficiently rough that the cylinders roll without slipping. The cylinders are connected at their centers by a spring of stiffness $\backslash ($ k $\backslash )$ and rest length $\backslash (\backslash$ell$>3$ r $\backslash ).$ The cylinders are released from rest under gravity at some initial angles of $\backslash (\backslash$alpha $\backslash )$ and $\backslash (\backslash$beta $\backslash ).$ Choose interesting initial angles so the simulation movie looks fun and choose $\backslash ($ k $\backslash )$ large enough to avoid collisions of the disks. Please include your code, some snapshots of your movie, and your derivation of the Euler$-$Lagrange equations. Here are some "strong" suggestions$/$hints:

$-$ Use $\backslash (\backslash$alpha $\backslash )$ and $\backslash (\backslash$beta $\backslash )$ as the two kinematic parameters that parametrize the configuration of the system.

$-$ To write your kinetic energy, make sure to recall how to relate the spins $\backslash (\backslash$omega$\_\{1\}\backslash )$ and $\backslash (\backslash$omega$\_\{2\}\backslash )$ to the corresponding values of $\backslash (\backslash$dot$\{\backslash$alpha$\}\backslash )$ and $\backslash (\backslash$dot$\{\backslash$beta$\}\backslash ).$ See problem $1$ above for a refresher.

$-$ Be aware of your sign conventions and stay consistent throughout.