## 2 Theoretical Background
### Hamiltonian Dynamics
\begin{aligned} \dot{G}&=\sum_{\alpha}\biggl{[}C_{ \alpha}^{\mathbb{Q}}\ \mathcal{A}_{\alpha}^{\mathbb{Q}}\ (G,H;L)\biggr{]}\frac{\partial\varphi_{\alpha}^{\mathbb{Q}}}{ \partial g}\sin(\varphi_{\alpha}^{\mathbb{Q}}\ )+\sum_{\gamma}\biggl{[}C_{\gamma}^{\odot} \mathcal{A}_{\gamma}^{\odot}(G,H;L)\biggr{]}\frac{\partial\varphi_{\gamma}^{ \odot}}{\partial g}\sin(\varphi_{\gamma}^{\odot}),\\ \dot{H}&=\sum_{\alpha}\biggl{[}C_{\alpha}^{\mathbb{Q} }\ \mathcal{A}_{\alpha}^{\mathbb{Q}}\ (G,H;L)\biggr{]}\frac{\partial\varphi_{\alpha}^{ \mathbb{Q}}}{\partial h}\sin(\varphi_{\alpha}^{\mathbb{Q}}\ )+\sum_{\gamma}\biggl{[}C_{\gamma}^{\odot} \mathcal{A}_{\gamma}^{\odot}(G,H;L)\biggr{]}\frac{\partial\varphi_{\gamma}^{ \odot}}{\partial h}\sin(\varphi_{\gamma}^{\odot}),\\ \dot{g}&=\frac{\partial\mathcal{H}_{J_{2}}}{\partial G }(G,H;L)+C_{0}^{\mathbb{Q}}\ \frac{\partial\mathcal{A}_{0}^{\mathbb{Q}}\ (G,H;L)}{ \partial G}+C_{0}^{\mathbb{Q}}\frac{\partial\mathcal{A}_{0}^{\mathbb{Q}}(G,H;L )}{\partial G}+\\ &\sum_{\alpha}\biggl{[}C_{\alpha}^{\mathbb{Q}}\ \frac{\partial\mathcal{A}_{ \alpha}^{\mathbb{Q}}}{\partial G}\ (G,H;L)\biggr{]}\cos(\varphi_{\alpha}^{\mathbb{Q}}\ )+\sum_{\gamma}\biggl{[}C_{\gamma}^{\odot} \frac{\partial\mathcal{A}_{\gamma}^{\odot}}{\partial G}(G,H;L)\biggr{]}\cos( \varphi_{\gamma}^{\odot}),\\ \dot{h}&=\frac{\partial\mathcal{H}_{J_{2}}}{\partial H }(G,H;L)+C_{0}^{\mathbb{Q}}\ \frac{\partial\mathcal{A}_{0}^{\mathbb{Q}}\ (G,H;L)}{ \partial H}+C_{0}^{\mathbb{Q}}\frac{\partial\mathcal{A}_{0}^{\mathbb{Q}}(G,H;L )}{\partial H}+\\ &\sum_{\alpha}\biggl{[}C_{\alpha}^{\mathbb{Q}}\ \frac{\partial\mathcal{A}_{ \alpha}^{\mathbb{Q}}}{\partial H}(G,H;L)\biggr{]}\cos(\varphi_{\alpha}^{\mathbb{ Q}}\ )+\sum_{\gamma}\biggl{[}C_{\gamma}^{\odot}\frac{\partial\mathcal{A}_{ \gamma}^{\odot}}{\partial H}(G,H;L)\biggr{]}\cos(\varphi_{\gamma}^{\odot}). \end{aligned} \right. \tag{11}\]  
Following, e.g., [15], the oblateness of the Earth does not produce any effect on the actions \(G\) and \(H\), but it causes a precession, or regression, of \(g\) and \(h\), which is usually used to approximate the evolution of the angles, as already mentioned before. From the last two equations of the system, Equation (11), we notice that the angles undergo secular drifts, caused by both the oblateness and the lunar and solar mean terms, and periodic effects, given by integrating the oscillating terms whose amplitude is proportional to the partial derivatives of the harmonic coefficients. Since the _Laplace radius_ is around 7.7 \(R_{\oplus}\)[17], higher than the Molinya one, and is the geocentric distance at which the order of magnitude of the precession caused by the lunisolar perturbation is equivalent to the one caused by the Earth's oblateness, the following approximation  
\[\left\{ \begin{aligned} \dot{g}&\approx\frac{\partial \mathcal{H}_{J_{2}}}{\partial G},\\ \dot{h}&\approx\frac{\partial\mathcal{H}_{J_{2}}}{ \partial H},\end{aligned} \right. \tag{12}\]  
is _usually_ both convenient and accurate enough. However, in the particular case of the Molinya dynamics, the orbits are critically inclined; thus, the third-body perturbation might not be necessarily negligible at least for \(g\), as confirmed by numerical experiments that will be presented in Section 3.1. From Equation (11), it is noteworthy that \(\dot{G}\), and thus the mean eccentricity, only depends on the harmonics showing \(\dot{g}\) in the argument. On the other hand, only harmonics depending on \(h\) affect \(\dot{H}\); therefore, the orbital inclination depends on the harmonics having \(g\) or \(h\) in the argument. Furthermore, the first two equations of the system (11) suggest that the larger the harmonic coefficient, the deeper the resulting fluctuations in \(G\) and \(H\). Following the approach presented in [10] for the dynamics in the main asteroid belt, we will analyze the role in the evolution of \(G\) and \(H\) also of the harmonic frequency.