= Solution
For the horizontal <generalized coordinate> $\eta$,
$$
\frac{\partial L}{\partial\dot\eta}=\dot\eta+\dot\theta\cos\theta,\qquad
\frac{\partial L}{\partial\eta}=-\frac km\eta.
$$
The corresponding <Euler-Lagrange equation> is
$$
\ddot\eta+\ddot\theta\cos\theta-\dot\theta^2\sin\theta+\frac km\eta=0.
$$
For $\theta$, the terms proportional to $\dot\eta\dot\theta\sin\theta$ cancel between $d(\partial L/\partial\dot\theta)/dt$ and $\partial L/\partial\theta$. Hence the other exact equation is
$$
\ddot\theta+\ddot\eta\cos\theta+\frac g\ell\sin\theta=0.
$$
Keeping only terms linear in small coordinates and their derivatives gives
$$
\boxed{\ddot\theta+\ddot\eta+\omega_0^2\theta=0,\qquad
\ddot\theta+\ddot\eta+\omega_1^2\eta=0,\qquad
\omega_0=\sqrt{\frac g\ell},\quad\omega_1=\sqrt{\frac km}.}
$$
The equilibrium is $\theta=\eta=0$. These characteristic frequencies label gravitational and spring restoring scales; they are not two independent <normal mode> frequencies of this massless-support system.
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