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Lagrangian for a Simple Pendulum

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A) Write the Lagrangian for a simple pendulum consisting of a mass m suspended at the end of a massless string of length l. Derive the equation of motion from the Euler-Lagrange equation, and solve for the motion in the small angle approximation.
B) Assume the massless string can stretch with a restoring force F=-k(r-ro) where ro is the unstretched length. Write the new Lagrangian and find the equations of motion.

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This solution contains step-by-step calculations and instructions to write the Lagrangian for a simple pendulum for different scenarios.

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