Chapter 9.5, Problem 12P

Find the Lagrangian and Lagrange’s equations for a simple pendulum (Problem 4 ) if the cord is replaced by a spring with spring constant $k$.

Hint: If the unstretched spring length is $r_0,$ and the polar coordinates of the mass $m$ are $(r,\theta ),$ the potential energy of the spring is $\frac{1}{2}k{\left(r-r_0\right)}^2$.