太阳-木星-土星-天王星系统平面模型的长期动力学; Kolmogorov和Nekhoroshev理论中的有效稳定性
Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories
摘要：
我们调查的长期稳定性的太阳-木星-土星-天王星系统考虑一个平面的长期模型，可以被视为一个主要的改进的方法首先介绍了拉格朗日。事实上，关于行星轨道的革命，我们改进了经典的圆形近似取代它的解决方案，是不变的，以两个顺序的质量，因此，我们调查的长期系统的稳定性，而小的值的偏心率。首先，我们显式地构造一个Kolmogorov标准形，以便找到一个不变的KAM环面，它很好地逼近长期轨道。最后，我们适应的方法是在分析部分的Nekhoroshev定理的基础上，以便表明，有一个附近的环面的估计稳定时间大于太阳系的寿命。这样一个邻域的大小，与天文观测的不确定性相比，大约要小十倍。
英文摘要：
We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.
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