= Elastic pendulum
{title2=$L=\tfrac12m(\dot r^2+r^2\dot\theta^2)-\tfrac12k(r-l)^2+mgr\cos\theta$}
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= Spring pendulum
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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.
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