## 5 Hamiltonians for Systems of \(N\) Ions
We denote  
\[D_{j}=\frac{1}{N}\sum_{\alpha=1}^{N}\frac{\partial}{\partial x_{\alpha j}}\,\ \ D_{ \alpha j}=\frac{\partial}{\partial x_{\alpha j}}-D_{j}\,\ \ \sum_{\alpha=1}^{N}D_{ \alpha j}=0. \tag{62}\]  
In addition  
\[\sum_{\alpha=1}^{N}\frac{\partial^{2}}{\partial x_{j}^{2}}=ND_{j}^{2}+\sum_{ \alpha=1}^{N}D_{\alpha j}^{2}. \tag{63}\]  
When \(d=3\) we denote by \(L_{\alpha 3}\) the projection of the angular momentum of the \(\alpha\) particle on axis 3. Then, the projections of the total angular momentum and of the relative motion angular momentum on axis 3, are labeled as \(L_{3}\) and \(L_{3}^{\prime}\) respectively, determined as  
\[ \begin{split}\sum_{\alpha=1}^{N}L_{\alpha 3}=L_{3}+L_{3}^{ \prime}\,\ \ L_{\alpha 3}=x_{\alpha 1}p_{\alpha 2}-x_{\alpha 2}p_{\alpha 1}\,\\ L_{3}=p_{1}D_{2}-p_{2}D_{1}\,\ L_{3}^{\prime}=\sum_{ \alpha=1}^{N}(y_{\alpha 1}\xi_{\alpha 2}-y_{\alpha 2}\xi_{\alpha 1})\.\end{split}  \tag{64}\]  
The Hamilton function assigned to a many body system of \(N\) charged particles of mass \(M\) and equal electric charge \(Q\), confined in a quadrupole combined (Paul and Penning)trap that displays axial (cylindrical) symmetry, in presence of a constant axial magnetic field \(B_{0}\), can be expressed as [1,36]  
\[H=\sum_{a=1}^{N}\left[\frac{1}{2M}\sum_{j=1}^{3}p_{aj}^{2}+\frac{K_{r}}{2}\left( x_{a1}^{2}+x_{a2}^{2}\right)+\frac{K_{a}}{2}x_{a3}^{2}-\frac{\omega_{c}}{2}L_{a3} \right]+W\,,\]  
with  
\[K_{r}=\frac{M\omega_{c}^{2}}{4}-2 Qc_{2}A(t)\,\ K_{a}=4 Qc_{2}A(t)\,\ \omega_{c}= \frac{QB_{0}}{M}\,\]  
where \(\omega_{c}\) is the cyclotron frequency characteristic to a Penning trap, \(c_{2}\) is a constant that depends on the trap geometry and \(A(t)\) represents a time periodical function [47]. The index \(r\) refers to radial motion while the index \(a\) refers to axial motion. We can also write \(H\) by adding the Hamilton function of the CM, \(H_{CM}\), and the Hamilton function associated to the ion relative motion \(H^{\prime}\):  
\[ \begin{split}& H=H_{CM}+H^{\prime}\,\\ & H_{CM}=\frac{1}{2 NM}\sum_{j=1}^{3}p_{j}^{2}+\frac{NK_{r}}{2} \Big{(}x_{1}^{2}+x_{2}^{2}\Big{)}+\frac{NK_{a}}{2}x_{3}^{2}-\frac{\omega_{c}} {2}L_{3}\,\\ & H^{\prime}=\sum_{a=1}^{N}\left[-\frac{\hbar^{2}}{2M}\sum_{j=1}^ {3}\xi_{aj}^{2}+\frac{K_{r}}{2}\Big{(}y_{a1}^{2}+y_{a2}^{2}\Big{)}+\frac{K_{a }}{2}y_{a3}^{2}\right]-\frac{\omega_{c}}{2}L_{3}^{\prime}+W\.\end{split}  \tag{65}\]  
Our results are in agreement with Ref. [22], where collective dynamical systems associated to the symplectic group are used to describe the axial and radial quantum Hamiltonians of the CM and of the relative ion motion. The space charge and its effect on the ion dynamics in case of a LIT is examined in Ref. [48], where the authors emphasize two distinguishable effects: (i) alteration of the specific ion oscillation frequency owing to variations of the trap potential, and (ii) for specific high charge density experimental conditions, the ions might perform as a single collective ensemble and exhibit dynamic frequency which is autonomous with respect to the number of ions. The model we suggest in this paper is appropriate to achieve a unitary approach aimed at generalizing the parameters for different types of 3D QIT. Further on, we apply this model to investigate the particular case of a combined Paul and Penning 3D QIT [49]. We consider \(W\) to be an interaction potential that is translation invariant (it only depends of \(y_{aj}\)). The ion distribution in the trap can be represented by means of numerical analysis and computer modeling [50,51], through the Hamilton function we provide  
\[ \begin{split} H_{sim}=\sum_{i=1}^{n}\frac{1}{2M}{p_{i}}^{2}+\sum _{i=1}^{n}\frac{M}{2}\Big{(}\omega_{1}^{2}x_{i}^{2}+\omega_{2}^{2}y_{i}^{2}+ \omega_{3}^{2}z_{i}^{2}\Big{)}\\ +\sum_{1\leq i\leq j\leq n}\frac{Q^{2}}{4\pi\epsilon_{0}}\frac{1 }{|\vec{r}_{i}-\vec{r}_{j}|}\,\end{split}  \tag{66}\]  
where the second term accounts for the effective electric potential of the 3D QIT and the third term is responsible for the Coulomb repulsive force.