= Solution
A <normal mode> has all coordinates following one fixed displacement pattern with a common time factor. Its <normal frequency> is that factor's oscillation frequency; <normal mode coordinates> are amplitudes in a basis of such patterns, making the linear equations independent. A zero-frequency mode can instead have constant velocity along a symmetry direction.
For the unit circle, each angular displacement equals its corresponding arc displacement. The <kinetic energy> and change in <potential energy> are
$$
T=\frac m2(\dot\theta_1^2+\dot\theta_2^2+\mu\dot\theta_3^2),\qquad V-V_0=\frac k2[(\theta_2-\theta_1)^2+(\theta_3-\theta_2)^2+(\theta_1-\theta_3)^2].
$$
Any common spring prestress changes only the constant $V_0$, because the sum of the three arc lengths is fixed. The <Euler Lagrange equations> give
$$
m\ddot\theta_1+k(2\theta_1-\theta_2-\theta_3)=0,\qquad m\ddot\theta_2+k(2\theta_2-\theta_1-\theta_3)=0,\qquad\mu m\ddot\theta_3+k(2\theta_3-\theta_1-\theta_2)=0.
$$
These are the <normal modes of three beads connected by springs on a circular hoop>, with third mass $M=\mu m$. For $\mu>0$, their three mode vectors and squared <angular frequencies> are
$$
\boxed{v_0=(1,1,1),\ \omega_0^2=0;\qquad v_A=(1,-1,0),\ \omega_A^2=\frac{3k}{m};\qquad v_S=(1,1,-2/\mu),\ \omega_S^2=\frac{k}{m}\left(1+\frac2\mu\right).}
$$
The zero mode rotates the entire configuration without stretching a spring. The antisymmetric mode moves the equal masses oppositely while leaving the third fixed. In the symmetric mode the equal masses move together against the third, with zero total <angular momentum>. These vectors are orthogonal in the mass inner product with matrix $m\operatorname{diag}(1,1,\mu)$.
Writing $\boldsymbol\theta=Q_0v_0+Q_Av_A+Q_Sv_S$ gives independent equations $\ddot Q_0=0$, $\ddot Q_A+\omega_A^2Q_A=0$, $\ddot Q_S+\omega_S^2Q_S=0$. In particular $Q_0=A+Bt$, retaining possible slow common rotation. For $\mu=1$, the two positive frequencies coincide at $\sqrt{3k/m}$; their two-dimensional space is all patterns with $\theta_1+\theta_2+\theta_3=0$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2-three-bead-modes.png]
{title=Rigid rotation and the two oscillatory patterns of three spring-coupled beads}
{description=Original illustrations for the example $\mu=2$. Arrows are tangent displacements at the equilibrium positions; spring extensions depend on differences of neighboring angular displacements.}
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