= Solution
For $q=(x_1,\eta_1,\eta_2)^T$, the <Euler-Lagrange equations> give
$$
m\ddot q+\mu Kq=0,\qquad K=\begin{pmatrix}19/3&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}.
$$
In components these are $m\ddot x_1=-\mu(19x_1/3-\eta_1)$, $m\ddot\eta_1=-\mu(2\eta_1-x_1-\eta_2)$ and $m\ddot\eta_2=-\mu(\eta_2-\eta_1)$. A <normal mode> has $q=v\cos(\omega t+\delta)$ and satisfies $Kv=(m\omega^2/\mu)v$. For <eigenvalue> $1/3$, the first and third rows give $\eta_1=6x_1$ and $\eta_2=3\eta_1/2$, and the second row then agrees. Hence
$$
\boxed{q=C(1,6,9)^T\cos\left(\sqrt{\frac\mu{3m}}t+\delta\right),\qquad P=2\pi\sqrt{\frac{3m}{\mu}}.}
$$
The particle positions are recovered by adding the equilibrium offsets to the last two entries. Their displacements are in phase with amplitude ratio $1:6:9$.
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