## 2 Theoretical Background
### Hamiltonian Dynamics
Let us consider a first approximation of the system (11) such that:  
* the actions are assumed as a constant, namely \[\mathcal{A}_{\alpha}^{\mathbb{Q}}\ (G,H;L)=\mathcal{A}_{\alpha}^{\mathbb{Q}}\, \mathcal{A}_{\gamma}^{\odot}(G,H;L)=\mathcal{A}_{\gamma}^{\odot},\]* the angles evolve linearly in time, namely \[ \begin{array}{ll}\varphi_{\alpha}^{\mathbb{(}}\ t)=\varphi_{\alpha,0}^{\mathbb{(} \mathbb{\zeta}}+\dot{\varphi}_{\alpha}^{\mathbb{(}\mathbb{\zeta}}\ t,\\ \end{array} \qquad\qquad\phi_{\gamma}^{\mathbb{(}\mathbb{\zeta}}(t)=\phi_{ \gamma,0}^{\mathbb{\zeta}}+\dot{\phi}_{\gamma}^{\mathbb{\zeta}}t,\end{array}\] being \(\varphi_{\alpha,0}^{\mathbb{(}\mathbb{\zeta}}\) and \(\phi_{\gamma,0}^{\mathbb{\zeta}}\) generic initial conditions and \(\phi_{\alpha}^{\mathbb{\zeta}}\) and \(\dot{\phi}_{\gamma}^{\mathbb{\zeta}}\) constants. Note that the last assumption holds especially in the neighborhood of a resonance. In this way, from (11), we obtain  
\[ \begin{cases}\dot{G}=\sum_{\alpha}C_{\alpha}^{\mathbb{\zeta}}\ \mathcal{A}_{\alpha}^{\mathbb{ \zeta}}\ \frac{\partial\varphi_{\alpha}^{\mathbb{(}\mathbb{\zeta}} }{\partial\mathbb{\zeta}}\sin(\varphi_{\alpha,0}^{\mathbb{(}\mathbb{\zeta}}+ \dot{\varphi}_{\alpha}^{\mathbb{(}\mathbb{\zeta}}\ t)+\sum_{\gamma}C_{\gamma}^ {\mathbb{\zeta}}A_{\gamma}^{\mathbb{\zeta}}\frac{\partial\phi_{\gamma}^{ \mathbb{\zeta}}}{\partial\mathbb{\zeta}}\sin(\phi_{\gamma,0}^{\mathbb{\zeta}}+ \dot{\phi}_{\gamma}^{\mathbb{\zeta}}t),\\ \dot{H}=\sum_{\alpha}C_{\alpha}^{\mathbb{\zeta}}\ \mathcal{A}_{\alpha}^{ \mathbb{\zeta}}\ \frac{\partial\varphi_{\alpha}^{\mathbb{(}\mathbb{\zeta}} }{\partial\mathbb{\zeta}}\sin(\varphi_{\alpha,0}^{\mathbb{\zeta}}+\dot{\varphi }_{\alpha}^{\mathbb{(}\mathbb{\zeta}}\ t)+\sum_{\gamma}C_{\gamma}^{\mathbb{ \zeta}}A_{\gamma}^{\mathbb{\zeta}}\frac{\partial\phi_{\gamma}^{\mathbb{\zeta}} }{\partial\mathbb{\zeta}}\sin(\phi_{\gamma,0}^{\mathbb{\zeta}}+\dot{\phi}_{ \gamma}^{\mathbb{\zeta}}t).\end{cases}  \tag{13}\]  
By integration, defining \(\Delta G=G(T)-G(0)\) and \(\Delta H=H(T)-H(0)\), we obtain  
\[ \begin{split}\Delta G=\sum_{\alpha}\frac{C_{\alpha}^{\mathbb{ \zeta}}\ \mathcal{A}_{\alpha}^{\mathbb{\zeta}}}{\dot{\varphi}_{\alpha}^{\mathbb{(} \mathbb{\zeta}}}\ \frac{\partial\varphi_{\alpha}^{\mathbb{(}\mathbb{\zeta}}}{ \partial\mathbb{\zeta}}\left[\cos(\varphi_{\alpha,0}^{\mathbb{(}\mathbb{\zeta}} )-\cos(\varphi_{\alpha}^{\mathbb{(}\mathbb{\zeta}}\ (T))\right]+\sum_{\gamma}\frac{C_{\gamma}^{\mathbb{\zeta}}A_{\gamma}^{ \mathbb{\zeta}}}{\dot{\phi}_{\gamma}^{\mathbb{\zeta}}}\frac{\partial\phi_{ \gamma}^{\mathbb{\zeta}}}{\partial\mathbb{\zeta}}\left[\cos(\varphi_{\gamma,0}^{ \mathbb{\zeta}})-\cos(\phi_{\gamma}^{\mathbb{\zeta}}(T)\right]\right],\\ \Delta H=\sum_{\alpha}\frac{C_{\alpha}^{\mathbb{\zeta}}\ \mathcal{A}_{\alpha}^{\mathbb{(} \mathbb{\zeta}}}{\dot{\varphi}_{\alpha}^{\mathbb{(}\mathbb{\zeta}}}\ \frac{\partial\varphi_{\alpha}^{\mathbb{(}\mathbb{\zeta}}}{ \partial\mathbb{\zeta}}\ \left[\cos(\varphi_{\alpha,0}^{\mathbb{\zeta}})-\cos(\varphi_{\alpha}^{ \mathbb{(}\mathbb{\zeta}}\ (T)\right])+\sum_{\gamma}\frac{C_{\gamma}^{\mathbb{\zeta}}A_{\gamma}^{ \mathbb{\zeta}}}{\dot{\phi}_{\gamma}^{\mathbb{\zeta}}}\frac{\partial\phi_{ \gamma}^{\mathbb{\zeta}}}{\partial\mathbb{\zeta}}\left[\cos(\phi_{\gamma,0}^{ \mathbb{\zeta}})-\cos(\phi_{\gamma}^{\mathbb{\zeta}}(T)\right]\right].\end{split}  \tag{14}\]  
In other words, the long-term variation of \(G\) and \(H\) depends on the _ratio_ between the amplitude of the harmonic coefficients and the corresponding frequency. In the following, this ratio will be denoted as \(R\). The larger the ratio, the deeper the long-term effects. Roughly speaking, a large _ratio_\(R\), that identifies a strong perturbing term, is due to a large amplitude or to a small frequency. In the first case, the term produces deep periodic oscillations of \(G\) and \(H\); in the latter, we are dealing with a resonant or near-resonant term leading to significant variation over the time.