= Solution
For the original <Lagrangian> of <1/c/solution>, differentiation with respect to the generalized <velocities> gives
$$
\boxed{p_{x_1}=m(\dot x_1-\nu y_1+R\Omega\sin\phi t),\qquad
p_{y_1}=m(\dot y_1+\nu x_1+R\Omega\cos\phi t).}
$$
In contrast, using $L'$ in the nonrotating case gives $p'_{x_1}=m\dot x_1$, $p'_{y_1}=m\dot y_1$; this is the expected <momentum> shift under adding a <total derivative>.
The <Euler-Lagrange equations> in the general cup frame are
$$
\ddot{\mathbf q}+2\nu\mathsf J\dot{\mathbf q}+(\omega^2-\nu^2)\mathbf q
=R\Omega^2(\cos\phi t,-\sin\phi t).
$$
These include the Coriolis and centrifugal terms when the cup axes really rotate. Set $z_q=x_1+iy_1$ to obtain
$$
\ddot z_q+2i\nu\dot z_q+(\omega^2-\nu^2)z_q=R\Omega^2e^{-i\phi t}.
$$
For $\omega^2\ne\Omega^2$, its complete solution is
$$
\boxed{z_q(t)=A e^{i(\omega-\nu)t}+B e^{-i(\omega+\nu)t}
+\frac{R\Omega^2}{\omega^2-\Omega^2}e^{-i\phi t},}
$$
where the two complex constants provide the four real initial data. The driving denominator is independent of cup-axis <rotation>: <rotation> changes the coordinate description, not the physical trap frequency.
In the requested special case $\nu=0$, an equivalent real form is
$$
\boxed{\begin{aligned}
x_1(t)&=a_x\cos\omega t+b_x\sin\omega t+D\cos\Omega t,\\
y_1(t)&=a_y\cos\omega t+b_y\sin\omega t+D\sin\Omega t,
\qquad D=\frac{R\Omega^2}{\omega^2-\Omega^2}.
\end{aligned}}
$$
For example $a_x=x_1(0)-D$, $b_x=\dot x_1(0)/\omega$, $a_y=y_1(0)$ and $b_y=[\dot y_1(0)-D\Omega]/\omega$ determine the constants.
At <resonance> $\omega^2=\Omega^2$ with nonzero $\Omega$, replace the particular term by
$$
\boxed{z_{q,\rm res}(t)=-\frac{iR\Omega}{2}\,t\,e^{-i\phi t}.}
$$
In the nonrotating cup frame this has components $(R\Omega t\sin\Omega t/2,-R\Omega t\cos\Omega t/2)$, plus the homogeneous solution. Secular growth eventually invalidates a finite cup or the neglected vertical <kinetic energy>. If the ride does not rotate, its forcing vanishes and ordinary free oscillations remain.
\Image[/past-exam-of-the-classical-mechanics-course-of-part-b-of-the-physics-course-of-the-university-of-oxford/2016-circular-trap-marble-response.png]
{title=Marble response in a nonrotating cup frame with a circularly accelerating origin: bounded off-resonant motion and secular resonant growth, starting at the cup center at rest}
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