## 5 Hamiltonians for Systems of \(N\) Ions  
We further apply our model to explore dynamical stability for systems consisting of \(N\) identical ions confined in a 3D QIT (Paul, Penning or combined traps) and show they can be studied locally in the neighbourhood of the minimum configurations that describe ordered structures (Coulomb or ion crystals [46]). Collective dynamics for many body systems confined in a 3D QIT that exhibits cylindrical (axial) symmetry is characterized in Refs. [1; 22]. We explore a system consisting of \(N\) ions in a space with \(d\) dimensions, labeled as \(\mathbb{R}^{d}\). The coordinates in the manifold of configurations \(\mathbb{R}^{d}\) are denoted by \(x_{aj}\,,\ \alpha=1,\ldots,N\,,\ j=1,\ldots,d\). In case of linear, planar or 3D (space) models, the number of corresponding dimensions is \(d=1\), \(d=2\) or \(d=3\), respectively. We will further introduce the kinetic energy \(T\), the linear potential energy \(U_{1}\), the 3D QIT potential energy \(U\), and the anharmonic trap potential \(V\):  
\[ \begin{split} T=\sum_{\alpha=1}^{N}\sum_{j=1}^{d}\frac{1}{2m_{ \alpha}}\ p_{aj}^{2}\,,\ \ U_{1}=\frac{1}{2}\sum_{\alpha=1}^{N}\sum_{j=1}^{d}\delta_{j}x_{aj}\,,\\ U=\frac{1}{2}\sum_{\alpha=1}^{N}\sum_{i,j=1}^{d}\kappa_{ij}x_{aj}^ {2}\,,\ \ V=\sum_{\alpha=1}^{N}v(\mathbf{x}_{\alpha},t)\,\end{split}  \tag{54}\]  
where \(m_{\alpha}\) is the mass of an ion labeled by \(\alpha,\mathbf{x}_{\alpha}=(x_{\alpha 1},\ldots,x_{\alpha d})\), while \(\delta_{j}\) and \(\kappa_{ij}\) represent functions that ultimately depend on time. The Hamilton function associated to the strongly coupled Coulomb system (SCCS) under investigation is  
\[H=T+U_{1}+U+V+W\,\]  
where \(W\) denotes the interaction potential between the ions. Under the assumption of equal ion masses we introduce \(d\) coordinates \(x_{j}\) of the center of mass (CM)  
\[x_{j}=\frac{1}{N}\sum_{\alpha=1}^{N}x_{\alpha j}\, \tag{55}\]  
and \(d(N-1)\) coordinates \(y_{aj}\) to account for the relative ion motion  
\[y_{aj}=x_{aj}-x_{j}\,\ \ \sum_{\alpha=1}^{N}y_{aj}=0. \tag{56}\]We also introduce \(d\) collective coordinates \(s_{j}\) and the collective coordinate \(s\), identified as  
\[s_{j}=\sum_{\alpha=1}^{N}y_{\alpha j}^{2}\,\ s=\sum_{\alpha=1}^{N}\sum_{j=1}^{d}y _{\alpha j}^{2}. \tag{57}\]  
Then  
\[\sum_{\alpha=1}^{N}x_{\alpha j}^{2}=Nx_{j}^{2}+\sum_{\alpha=1}^{N}y_{\alpha j}^ {2}. \tag{58}\]  
\[s_{j}=\frac{1}{2N}\sum_{\alpha,\beta=1}^{N}\left(x_{\alpha j}-x_{\beta j} \right)^{2}\,\ \ s=\frac{1}{2N}\sum_{\alpha,\beta=1}^{N}\sum_{j=1}^{d}\left(x_{\alpha j}-x_{ \beta j}\right)^{2}. \tag{59}\]  
Equation (59) shows \(s\) to symbolize the squared distance measured between the origin (fixed in the CM) and the point that designates the system of \(N\) ions in the manifold of configurations. The relation \(s=s_{0}\), with \(s_{0}>0\) constant, establishes a sphere of radius \(\sqrt{s_{0}}\) whose centre is located in the origin (of the configurations manifold). When investigating ordered structures of \(N\) ions, the trajectory is restricted within a neighbourhood \(\|s-s_{0}\|<\varepsilon\) of this sphere, with \(\varepsilon\) sufficiently small. At the same time, the collective variable \(s\) can be also regarded as a dispersion:  
\[s=\sum_{\alpha=1}^{N}\sum_{j=1}^{d}\left(x_{\alpha j}^{2}-x_{j}^{2}\right). \tag{60}\]  
We now submit \(p_{\alpha j}\) moments associated to the coordinates \(x_{\alpha j}\). We also introduce \(d\) moments \(p_{j}\) of the center of mass (CM) and \(d(N-1)\) moments \(\xi_{\alpha j}\) of the relative ion motion defined as  
\[p_{j}=\frac{1}{N}\sum_{\alpha=1}^{N}p_{\alpha j}\,\ \ \xi_{\alpha j}=p_{ \alpha j}-p_{j}\,\ \ \sum_{\alpha=1}^{N}\xi_{\alpha j}=0\, \tag{61}\]  
with \(p_{\alpha j}=-i\hbar\left(\partial/\partial x_{\alpha j}\right)\).