论文标题
多谐波式汉密尔顿模型，并应用于一阶共振
Multi-harmonic Hamiltonian models with applications to first-order resonances
论文作者
论文摘要
在这项工作中，制定了两个用于平均运动共振的多谐波式哈密顿模型，并讨论了它们在一阶共振上的应用。对于$ k_p $：$ k $ resonance，通常的关键参数$φ=kλ -k_pλ_p +（k_p -k）\ varpi $被视为第一个模型中的共振角度，而第二款模型则以新的关键参数为特征$σ=φ/ k_p $。基于规范转换，制定了与这两个模型相关的共鸣的哈密顿量。发现与第一个模型相比，第二个汉密尔顿模型具有两个优点：（a）提供相肖像和庞加莱段之间的直接对应关系，以及（b）呈现新的相位相结构，其中零含量份量是一个可见的鞍点。然后，将第二个汉密尔顿模型应用于一阶内部和外部共振，包括2：1、3：2、4：3、2：3和3：4共振。这些一阶共振的相位结构将进行详细讨论，然后通过分析识别库中心和相关的谐振宽度。仿真结果表明，随着偏心率接近零，偏心分离率偏离标称共振位置，库中心中心与名义共鸣位置不同，尤其是在内部和外部（第一阶）的谐振时，共振分离均不会消失。
In this work, two multi-harmonic Hamiltonian models for mean motion resonances are formulated and their applications to first-order resonances are discussed. For the $k_p$:$k$ resonance, the usual critical argument $φ= k λ- k_p λ_p + (k_p - k) \varpi$ is taken as the resonant angle in the first model, while the second model is characterized by a new critical argument $σ= φ/ k_p$. Based on canonical transformations, the resonant Hamiltonians associated with these two models are formulated. It is found that the second Hamiltonian model holds two advantages in comparison to the first model: (a) providing a direct correspondence between phase portraits and Poincaré sections, and (b) presenting new phase-phase structures where the zero-eccentricity point is a visible saddle point. Then, the second Hamiltonian model is applied to the first-order inner and outer resonances, including the 2:1, 3:2, 4:3, 2:3 and 3:4 resonances. The phase-space structures of these first-order resonances are discussed in detail and then the libration centers and associated resonant widths are identified analytically. Simulation results show that there are pericentric and apocentric libration zones where the libration centers diverge away from the nominal resonance location as the eccentricity approaches zero and, in particular, the resonance separatrices do not vanish at arbitrary eccentricities for both the inner and outer (first-order) resonances.
