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

我们根据经典力学中运动的产生者Liouville算子的特征值问题分析了一个不可积系统。我们主要研究由小行星、太阳和木星组成的受限三体问题，这是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，我们可以分析小行星分布中的共振效应。我们得到了在木星与小行星的周期之比用简单整数表示的一个共振点处的摄动柳维良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，只有几千年的量级。

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.