# Dynamics of Polar Resonances and Their Effects on Kozai-Lidov Mechanism
## 6 Conclusions  
In this paper, we study the dynamical maps of polar resonance and its effect on the Kozai-Lidov mechanism in the context of CRTBP using numerical and semi-analytical methods by mutual authenticating. Previous studies on resonance paid little attention to the perpendicular orbits with inclination \(i=90^{\circ}\). We introduce the action-angle variables for polar resonances and deduce proper canonical transformations. We can calculate the Hamiltonian by averaging over the fast angles and plot the level curves of the Hamiltonian on a \(e-\omega\) space, which can be used to understand the Kozai-Lidov dynamics inside polar resonance.  
Then, we analyze the polar resonant dynamics of interior and exterior resonances, taking the \(2/1\) and \(2/3\) resonances as examples. The resonant libration centers and width are studied for each case. The most apparent difference in dynamics of \(2/1\) and \(2/3\) resonance is their libration centers. As shown in Figures 4 and 7, the dominated libration centers are always at \(\sigma=0^{\circ}\) for \(2/1\) resonance, while the centers around \(\sigma=180^{\circ}\) appear in \(2/3\) resonance. The determination of the resonance center provides the basis for the following semi-analytical work. The Kozai-Lidov dynamics inside polar resonance are depicted on the \(e-\omega\) space as shown in Figures 5 and 8. Evolutions of several representative particles are also depicted and correspond very well with level curves with Hamiltonian. We confirm that the moderate libration amplitude of critical angle \(\sigma_{\text{amp}}\) has no pronounced effect on \(e-\omega\) portraits. The lifetime of 5000 particles in resonance are investigated by numerical integrations in Figures 1 and 8. All of the dead particles are due to their eccentricities rising to \(1\). The \(e-\omega\) spaces can explain the numerical performances for interior and exterior polar resonances, which demonstrates that the \(e-\omega\) spaces represent the real Kozai-Lidov dynamics inside polar resonances. These phase-space portraits of the Kozai-Lidov mechanism can be used to investigate the long-term evolution of polar resonant particles. As a supplement, we study the pure secular dynamics outside polar \(2/1\) resonance, and demonstrate that we cannot ignore the effect of polar resonance on secular dynamics. In this way, we can also study other resonant secular dynamics. Our work can be used to understand the dynamics of minor bodies that are in polar resonance with the planet. It may also help to look for objects with polar orbits in the resonant region of the planet. We will investigate the dynamics of exterior polar \(1/n\) resonance where the asymmetric librations may exist in our next work.  
Conceptualization, M.L. and S.G.; methodology, M.L.; resources, S.G.; data curation, M.L.; writing--original draft preparation, M.L.; writing--review and editing, S.G. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China (Grant No. 11772167). We appreciate very much the suggestions of the editor and reviewer. These suggestions will benefit the improvement of this paper and our future research. was supported by the National Natural Science Foundation of China (Grant No. 11772167). The authors declare that they have no conflict of interest.