Analysis of non-integrable system by the eigenvalue problem of the Liouvillian in classical mechanics

我们已经分析了一个不可积系统的特征值问题的刘维算子（刘维），这是在经典力学中的运动的发电机。我们专注于限制三体问题，其中包括一个小行星，太阳和木星作为一个典型的不可积系统。利用量子力学中发展起来的简并微扰理论，可以分析小行星分布中的共振效应。我们得到了在木星周期与小行星周期之比用简单整数表示的共振点处的摄动Liouvillian本征值问题的近似解。这些解表明，小行星分布中条纹图案形成的时间尺度相当短，大约为千年。

We have analyzed a non-integrable system in terms of the eigenvalue problem of the Liouville operator (the Liouvillian) that is the generator of motion in classical mechanics. We focus on the restricted three-body problem that consists of an asteroid, Sun, and Jupiter as one of the typical non-integrable systems. By using the degenerate perturbation theory which has been developed in quantum mechanics, we can analyze the resonance effect in the distribution of asteroids. We have obtained the approximate solutions of the eigenvalue problem of the perturbed Liouvillian at one of the resonance points where the ratios of the period of Jupiter to that of an asteroid are expressed by simple integers. These solutions indicate that the time scale of the formation of the striped pattern in the asteroid distribution is rather short in the order of thousand years.