Elastic pendulum ($L=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2)-\frac{1}{2}k(r-l{)}^2+mgr\cos \theta$)

A point mass attached to a massless straight spring can change its radial distance $r$ from a fixed support while swinging in a vertical plane. Its lower equilibrium length is $r_0=l+mg/k$. Linearizing the Euler-Lagrange equations there gives independent radial and angular normal modes with squared angular frequencies $k/m$ and $g/r_0$. The formal upper stationary state has $r=l-mg/k$ and exists in positive-length coordinates only when $l>mg/k$; its angular mode is unstable.