# Dynamics of Polar Resonances and Their Effects on Kozai-Lidov Mechanism
## 1 Introduction  
Mean motion resonances (MMRs) occur between two or more objects when their orbital periods are commensurable. MMRs with planets play a fundamental role in dynamics in the Solar System [1; 2; 3; 4; 5; 6; 7; 8]. [PERSON] et al. [9] provided a good review of resonance research up to the beginning of the 21 st century. Resonances are an essential mechanism in the dynamics of asteroids, Centaurs, trans-Neptunian objects (TNOs), satellite systems, and the planetary system [4; 10; 11; 12; 13; 14; 15]. The resonant dynamics at an arbitrary inclination, including retrograde motions, have attracted a lot of attention with the discovery of the more highly inclined objects. [PERSON] and [PERSON] [16; 17] numerically studied the resonance capture at an arbitrary inclination in the three-body problem. [PERSON] and [PERSON] [18] identified a set of asteroids whose inclination \(i>140^{\circ}\) is in retrograde mean motion resonance with giant planets. The research on the co-orbital dynamics in three dimensions after demonstrating the first retrograde co-orbital asteroid 2015 BZ509 has become a hot topic in recent years [19; 20; 21; 22; 23; 24]. [PERSON] et al. [25] systematically presented the retrograde resonant configurations of the asteroids in the Solar System. Recently, [PERSON] [26] analytically gave the expression of the half-width of the MMRs in spatial motion, and an analogue expression was then derived from the Hamiltonian dynamics by [PERSON] [27]. Resonant width and libration in three dimensions are studied by [PERSON] [15] and [PERSON] and [PERSON] [28], and they chose different approaches for measuring the strength of resonances. The Kozai-Lidov mechanism of minor bodies is a basic astrophysical phenomenon that has been studied for years [29; 30]. In the model considered by [PERSON] [30], the vertical angular momentum \(\sqrt{1-e^{2}}\text{cos}i\) is an additional integral of motion. The corresponding representative performance is the periodic variation in the inclination and eccentricity of an orbiting body [31]. Nowadays, the Kozai-Lidov mechanism is considered to play a critical role in the evolution of both the Solar System and extrasolar systems [31; 32; 33; 34; 35; 36]. The semi-analytical averaging model is a practical approach to understand the Kozai-Lidov mechanism of minor bodies [37; 38; 39; 13]. Based on the study of the secular hierarchical triple system by [PERSON] [30], the eccentric Kozai-Lidov mechanism (EKM) where the perturber is in an eccentric orbit has been studied [40; 41]. The most distinctive phenomenon is that the orbit can flip its orientation, form prograde (\(i<90^{\circ}\)) to retrograde (\(i>90^{\circ}\)), or vice versa [42; 43; 44; 45]. However, if we consider both the Kozai-Lidov mechanism and mean motion resonances simultaneously, the dynamics become much more different and challenging to fathom [21]. As [PERSON] [13] described, secular dynamics inside mean motion resonances would be one of the most implicated problems in celestial mechanics. [PERSON] [46] adapted the secular model from [PERSON] [30] to study the secular perturbation of resonant asteroids under the assumption that the critical arguments are fixed at stable libration centers [47; 48]. Afterward, that model was adapted to analyze the secular dynamics of resonance [49; 50; 51]. Secular resonances in mean motion commensurabilities are investigated in \(2/1,3/2\), etc. [52; 53]. [PERSON] et al. [54] explored the resonant and secular dynamics of KBOs with the canonical transformation. [PERSON] et al. [9] considered the Kozai-Lidov dynamics inside the prograde co-orbital resonance. [PERSON] and [PERSON] [55] studied the topological structure of the plane of the eccentricity and the argument of the perihelion analytically. [PERSON] et al. [56] investigated the secular process in \(1/1\) resonance using numerical averaging over the \"fast\" and \"semi-fast\" motion. [PERSON] and [PERSON] [57] constructed the Lidov-Kozai configuration for planets in co-orbital 3D motion. [PERSON] et al.