Consider a simple pendulum of mass \(m_{2}\) and arm length \(l\) having its pivot on a point of support of mass \(m_{1}\) that is free to move horizontally on a frictionless rail.

(a) Find the Lagrangian of the system in terms of the two degrees of freedom \(x\) and \(\theta\) shown on the figure. Do NOT assume small displacements.

(b) Identify two symmetries and the two corresponding conservation laws. Write two first-order differential equations that describe the dynamics of the two degrees of freedom \(x\) and \(\theta\). Correspondingly, write a single nasty integral for \(\theta(t)\).

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9 X m1 0 Sliding pendulum. M2