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Coupled oscillations Consider a system of two pendulums connected by a spring (see figure) 1. Derive a Lagrangian of this system using angles θ1 and θ2 as generalized coordinates. Let the distance between the suspension points of the pendulums to be l , and do not forget to include the energy of the spring. 12 points 2. Define generalized momentums for each generalized coordinate and derive an expression for the Hamiltonian of this system. 8 points 3. Write down Lagrangian and Hamiltonian equations of motion. 8 points 4. Considering only the case of small oscillations ( 1,2 θ 1 ) reduce the derived equations to that of coupled harmonic oscillators. 5 points 5. Find normal frequencies of the system 15 points 6. Find normal modes of the system. 20 points 7. Solve the equations of motion assuming the following initial conditions: 1 0 2 1,2 θ θθ θ = = = , 0, 0  and find the time dependence of each angle. 22 points 8. Compute the energy of each pendulum individually and describe its time dependence. At which time instances all the energy is concentrated in only one of the pendulums? Find these instances for each pendulum 10 points.

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