Find the Hamiltonian for the system described in Exercise 1 and obtain Hamilton’s equation...

Find the Hamiltonian for the system described in Exercise 1 and obtain Hamilton’s equations of motion for the system. Use both the direct and the matrix approach in finding the Hamiltonian.

Exercise 1

A uniform bar of mass M and length 2l is suspended from one end by a spring of force constant k. The bar can swing freely only in one vertical plane, and the spring is constrained to move only in the vertical direction. Set up the equations of motion in the Lagrangian formulation.