## 3 Characterization of Invariant Relative Orbits
### Characterization of \(J_{n}\)-Invariant Relative Trajectories  
The transformation \(\mathcal{T}_{7}\) produces the following form of the Hamiltonian function:  
\[\hat{H}=\alpha w_{1}z_{1}+\frac{\beta_{1}}{2}\Big{(}w_{2}^{2}+z_{2}^{2}\Big{)}+ \frac{\beta_{2}}{2}\Big{(}w_{3}^{2}+z_{3}^{2}\Big{)} \tag{31}\]  
The use of Hamilton Equation (3) shows that the dynamics on the three planes \((w_{i},z_{i})\) are not coupled and, because the Hamiltonian function is constant (see Appendix A), the three terms of Equation (31) are also constant.  
The trajectories associated with the first term, and thus associated with the saddle type of equilibrium, are hyperbolas of equations  
\[\begin{cases}w_{1}(t)=w_{1}(t_{0})e^{\alpha t}\\ z_{1}(t)=z_{1}(t_{0})e^{-\alpha t}\end{cases} \tag{32}\]  
where \(t_{0}\) indicates the value of the state variables at a specific time. Equation (32) is associated with trajectories that can diverge from \((z_{1}(t_{0})=0)\), converge to \((w_{1}(t_{0})=0)\), or oscillate indefinitely in time \((w_{1}(t_{0})=z_{1}(t_{0})=0)\) in the neighborhood of \(\mathbf{x}^{\star}\). In fact, the condition \(w_{1}(t_{0})=z_{1}(t_{0})=0\) characterizes relative trajectories which are stably bounded (do not drift in time) also in the presence of the zonal harmonic perturbation.  
Before validating the results through numerical integration, presented in Section 4, a preliminary proof of the model consists in verifying that it produces the well-known boundedness conditions for the unperturbed case, which can be derived from the HCW equations [11]  
\[\begin{cases}v_{y}=-2x\\ v_{x}=\frac{y}{2}\end{cases} \tag{33}\]  
This can be easily done by setting to zero the \(K_{ii}\) terms in Equation (25), applying the Siegel-Moser transformation defined in Section 3.2 and making the condition \(w_{1}=z_{1}=0\) explicit:  
\[\begin{cases}x+p_{y}=1\\ y+p_{x}=0\end{cases} \tag{34}\]  
Introducing the expressions of \(p_{x}\) and \(p_{y}\) from Equation (4) shows that two sets of Equations (33)-(34) are equal.  
The last two terms of Equation (31) are harmonic oscillators associated with the center types of equilibrium. Recalling the properties in Section 3.2, it is possible to conclude that the amplitude of the in-plane oscillations is proportional to \(|w_{2}|\) and, equivalently, that of the out-of-plane oscillations is proportional to \(|w_{3}|\)[33].  
It is worth summarizing here the main results obtained in Section 3:  
* Applying the canonical transformations \(\mathcal{T}_{1}\)-\(\mathcal{T}_{6}\), the dynamics equations of relative motion in the full zonal problem are converted to a form equivalent to that of the unperturbed case plus negligible second-order terms. Trajectories designed in this framework are invariant with respect to the effects of the zonal harmonic perturbation.
* A last canonical transformation \(\mathcal{T}_{7}\) produces a simplified representation of the dynamics, which can provide a compact characterization of invariant relative trajectories.
* Bounded relative trajectories evolving around \(\mathbf{x}^{\star}\) (i.e., the chief satellite) are quasiperiodic orbits characterized by \(w_{1}(t_{0})=z_{1}(t_{0})=0\).
* The amplitude of the in-plane oscillations is proportional to \(|w_{2}|\) (or equivalently \(|z_{2}|\)).
* The amplitude of the out-of-plane oscillations is proportional to \(|w_{3}|\) (or equivalently \(|z_{3}|\)).