(a) Write down the Lagrangian for a simple pendulum constrained to move in a single vertical plane. Find from it the equation of motion and show that for small displacements from equilibrium the pendulum performs simple harmonic motion. (b) Consider a particle of mass m moving in one dimension under a force with the potential U (x ) = k(2x^3-5x^2+4x), where the constant k > 0. Show that the point x = 1 corresponds to a stable equilibrium position of the particle. Find the frequency of a small amplitude oscillation of the particle about this equilibrium position.
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