We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on the simplest possible dynamical structure provided by the Keplerian problem perturbed by the resonant coupling truncated at second order in the eccentricities. The reduced Hamiltonian, constructed by a transformation to resonant coordinates, is then submitted to a suitable ordering of the terms and to the study of its equilibria. Henceforth, resonant normal forms are computed. The main result is the identification of two different classes of equilibria. In the first class, only one kind of stable equilibrium is present: the paradigmatic case is that of the Galilean system. In the second class, three kinds of stable equilibria are possible and at least one of them is characterised by a high value of the forced eccentricity for the ‘first planet’: here, the paradigmatic case is the exo-planetary system GJ-876, in which the combination of libration centres admits triple conjunctions otherwise not possible in the Galilean case. The normal form obtained by averaging with respect to the free eccentricity oscillations describes the libration of the Laplace argument for arbitrary amplitudes and allows us to determine the libration width of the resonance. The agreement of the analytic predictions with the numerical integration of the toy models is very good.

我们描述了锁定在拉普拉斯共振中的系统的综合模型。该框架基于由开普勒问题提供的最简单的可能的动力学结构，该结构受偏心率二阶截断的共振耦合的干扰。通过转换为共振坐标构造的简化哈密顿量，然后服从这些项的适当排序，并对其平衡性进行研究。此后，计算共振法线形式。主要结果是确定两种不同类别的平衡。在第一类中，仅存在一种稳定的平衡：伽利略系统的典型情况。在第二类中，可能存在三种稳定的平衡，并且其中至少一种具有“第一行星”的强制偏心率值高的特征：这里的典型案例是系外行星系统GJ-876，其中解放中心的组合允许三重合取，否则在加利利案中是不可能的。通过对自由偏心率振荡求平均值而得到的法线形式描述了Laplace自变量对于任意振幅的自由度，并允许我们确定共振的自由度宽度。分析预测与玩具模型的数值积分的一致性非常好。通过对自由偏心率振荡求平均值而得到的法线形式描述了Laplace自变量对于任意振幅的自由度，并允许我们确定共振的自由度宽度。分析预测与玩具模型的数值积分的一致性非常好。通过对自由偏心率振荡求平均值而得到的法线形式描述了Laplace自变量对于任意振幅的自由度，并允许我们确定共振的自由度宽度。分析预测与玩具模型的数值积分的一致性非常好。