= Solution
There are \b[two degrees of freedom], conveniently the azimuthal angles $\theta_1,\theta_2$ on the two circles. The positions are $(r_1\cos\theta_1,r_1\sin\theta_1,0)$ and $(r_2\cos\theta_2,r_2\sin\theta_2,c)$. Their squared separation is $d^2=r_1^2+r_2^2+c^2-2r_1r_2\cos(\theta_1-\theta_2)$. Thus the <Lagrangian> is
$$
\boxed{L=\frac12m_1r_1^2\dot\theta_1^2+\frac12m_2r_2^2\dot\theta_2^2-\frac12\omega^2\left[r_1^2+r_2^2+c^2-2r_1r_2\cos(\theta_1-\theta_2)\right].}
$$
The constant terms may be omitted from the equations of motion. In this notation $\omega^2$ is the stated spring coefficient, with the units needed for <potential energy>; it is not automatically an angular frequency for arbitrary <masses>.
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