 Solution

There are 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 $d^2=r_1^2+r_2^2+c^2-2r_1{r}_2\cos ({\theta }_1-{\theta }_2)$. Thus the Lagrangian isThe 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.