= Solution
The quadratic <Lagrangian> is diagonal in $x,y,\theta$, so the <Euler-Lagrange equations> give two pure translations and one pure rotation. Their <angular frequencies> are
$$
\boxed{\omega_x^2=\omega_y^2=\frac{3k}{2m}\left(1+\frac{l-l_0}{l}\right),\qquad
\omega_\theta^2=\frac{6k(l-l_0)(a+l)}{mal}.}
$$
A horizontal translation has $(x,y,\theta)=(A\cos\omega_xt,0,0)$; a vertical translation has $(0,A\cos\omega_yt,0)$; and the rotational mode has $(0,0,A\cos\omega_\theta t)$. Any fixed-direction translation is another choice in the same two-dimensional degenerate <normal mode> space. The rotation keeps the centre fixed.
\Image[/radially-sprung-disk-normal-modes.png]
{title=Two degenerate translations and a separate rotation of a disk on symmetric radial springs}
{description=Original schematic of the three <normal modes>; arrows indicate the independent motions, and the dashed disk marks the equilibrium position.}
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