Birkhoff正规形的计算机辅助估计
Computer-assisted estimates for Birkhoff normal forms
摘要：
Birkhoff规范形是保证Hamilton系统在椭圆平衡点附近“有效稳定”的常用形式。从理论的角度来看，这意味着最终的扩散可以有界的时间间隔是指数大的相对于从这样的平衡点的初始条件的距离的倒数。在这里，我们专注于一种方法，是适合于实际应用：我们扩展了一个相当经典的计划的Birkhoff规范形式的任何有限的顺序和他们的granders的估计。这是为了提供明确的下限的稳定时间（这是有效的初始条件，在一个固定的开放球），通过使用一个完全严格的计算机辅助程序。我们将我们的方法应用于天体力学中广泛研究的两个简单的环境：Hénon-Heiles模型和圆平面限制性三体问题。在后一种情况下，我们适应我们的计划估计覆盖共振Birkhoff正常形式的情况下，并在一些具体的特洛伊小行星的运动模型，我们表明，它可以更有利的相对于通常的非共振的。
英文摘要：
Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the Hénon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.
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