in $\Gamma$ is wrapped around an invariant 6-torus $\mathbb T^6$. By the KAM (Kolmogorov-Arnold-Moser) theorem [@Goldstein], the torus is stable under the perturbations induced by the fluctuations of $X$ and $x$, and it is slowly deformed with the global evolutions of $X(t)$ and of $x(t)$.\
Now we consider the case $N>1$. By a similar approach, the Jacobian matrix at $0$ of the dynamical system governing the quantum fluctuations is $$\partial F_{|(|\delta x\rangle,|\delta p \rangle) = 0} = \left(\begin{array}{cc} S & 0 \\ 0 & S \end{array} \right), \qquad S = \left(\begin{array}{cc} 0 & -I_N \\ {\mathrm{id}}_{3N} & 0 \end{array} \right)$$ with the $\mathfrak M_{N \times N}(\mathbb C)$-valued “inertia matrix” $$I_N = \left(\begin{array}{ccc} Y_2^2+Y_3^2 & -Y_2Y_1+[Y_1,Y_2] & -Y_3Y_1 + [Y_1,Y_3] \\ -Y_1Y_2 + [Y_2,Y_1] & Y_1^2+Y_3^2 & -Y_3Y_2 + [Y_2,Y_3] \\ -Y_1 Y_3 + [Y_3,Y_1] & -Y_2 Y_3 + [Y_3,Y_2] & Y_1^2+Y_2^2 \end{array} \right)$$ Let $|\delta x_i \rangle = \delta x_i |u_i \rangle$ be a decomposition of the fluctuation vectors with $\delta x_i$ being the complex amplitude of the quantum fluctuations and $|u_i \rangle$ being the “polarisation” of the fluctuations (with $\|u_i(t=0)\|=1$). We set that $\delta x_i$ obeys to the same equation that in the case $N=1$ (with an abelianization of 