A uniform thin rod of length \(\ell\) and mass \(m\) is suspended from fixed points \(A\) and \(B\) by two identical massless springs with zero natural length and spring constant \(k\). At equilibrium (as shown in the figure), the angle between the springs and the vertical is \(\phi\) and the length of the springs is \(a\). All motion is restricted to this vertical plane.

(a) Construct the Lagrangian and obtain Euler-Lagrange equations.

(b) Find the normal modes of small oscillations of the system about equilibrium and their angular frequencies.

(c) Construct the Hamiltonian, obtain Hamilton's equations and show equivalence to the EulerLagrange equations.

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