= Solution
A <normal mode> is a motion in which all generalized coordinates oscillate at one <frequency> with fixed <amplitude> ratios. Small <oscillations> about <stable equilibrium> in a time-independent conservative system give $\mathsf M\ddot{\boldsymbol\theta}+\mathsf K\boldsymbol\theta=0$, with positive <mass matrix> and symmetric <stiffness matrix>, allowing a generalized normal-mode eigenproblem.
For the <three-to-one mass double-pendulum modes>, the angles are absolute displacements from vertical, as printed. To quadratic order,
$$
K=\frac{m\ell^2}{2}(4\dot\theta_1^2+2\dot\theta_1\dot\theta_2+\dot\theta_2^2),\qquad
U=\frac{mg\ell}{2}(4\theta_1^2+\theta_2^2).
$$
For $\boldsymbol\theta=\mathbf A\cos\omega t$ and $q=\omega^2\ell/g$, the determinant is
$$
\det\begin{pmatrix}4-4q&-q\\-q&1-q\end{pmatrix}
=3q^2-8q+4=(3q-2)(q-2).
$$
Thus
$$
\boxed{\omega_-^2=\frac{2g}{3\ell},\quad(\theta_1,\theta_2)\propto(1,2);
\qquad\omega_+^2=\frac{2g}{\ell},\quad(\theta_1,\theta_2)\propto(1,-2).}
$$
The low <mode> has the strings leaning together, with the lower angle twice as large; the high <mode> has opposite angular signs. Both reverse together after half their respective periods.
For the specified release, decompose $(\alpha,\alpha)=(3\alpha/4)(1,2)+(\alpha/4)(1,-2)$. Therefore
$$
\theta_1=\frac\alpha4(3\cos\omega_-t+\cos\omega_+t),\qquad
\theta_2=\frac\alpha2(3\cos\omega_-t-\cos\omega_+t),
$$
and the lower <mass>'s horizontal displacement is
$$
\boxed{x_m\simeq\ell(\theta_1+\theta_2)
=\frac{\alpha\ell}{4}(9\cos\omega_-t-\cos\omega_+t),\qquad
\sup|x_m|=\frac52|\alpha|\ell.}
$$
This is the <supremum of two incommensurate harmonic modes>, conventionally described as the maximum-amplitude estimate. Strictly, $\omega_+/\omega_-=\sqrt3$ is irrational, so the opposite cosine extrema are approached arbitrarily closely but never coincide at a finite time. The quoted value is a <supremum> of the ideal linearized motion.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1-gpp2014-double-pendulum.png]
{title=Original normal-mode motion sketches and the lower-mass displacement after equal-angle release, with the limiting amplitude bound marked}
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