$5.(30)(1)$ There is a pendulum that made of a massless straight spring connected to a fixed ceiling with rod length $\backslash ($ l $\backslash )$ and mass $\backslash ($ m $\backslash ).$ The rod length $\backslash ($ l $\backslash )$ is not fixed but variable in this pendulum with a spring constant $\backslash ($ k $\backslash ).($equilibrium length is $\backslash ($ l$\_\{0\}\backslash ))$ We can neglect the mass of the rod. The pendulum rod is made of a spring with spring constant of $\backslash ($ k $\backslash )$ and the unstretched length is $\backslash ($ l $\backslash ).$ The spring remains straight while the pendulum is oscillating. Let the length of the pendulum rod, $\backslash ($ l $\backslash ),$ and the potential energy for the pendulum spring is $\backslash (\backslash$frac$\{1\}\{2\}$ k$\backslash$left$($l$-$l$\_\{0\}\backslash$right$)^\{2\}\backslash ).$

$(1-1)$ Find the Lagrangian of this system.

$(1-2)$ Find the equations of motion in this system.

$(2)$ This pendulum with a massless spring is attached to the ceiling inside a train that is accelerating into the right direction with the acceleration of $\backslash ($ a $\backslash ).$ For your convenience, let the initial velocity of this train be zero $\backslash (\backslash$left$($v$\_\{0\}=0\backslash$right$)\backslash ).$

$(2-1)$ Find the Lagrangian of this system.

$(2-2)$ Find the equations of motion of this system.