Pub Date : 2024-09-09DOI: 10.1088/1361-6544/ad72c6
Massimiliano Guzzo
A gravitational close encounter of a small body with a planet may produce a substantial change of its orbital parameters which can be studied using the circular restricted three-body problem. In this paper we provide parametric representations of the fast close encounters with the secondary body of the planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the consequence of a remarkable factorisation of the Birkhoff normal forms of the Hamiltonian of the problem represented with the Levi–Civita regularisation. The parameterisations are computed using two different sequences of Birkhoff normalisations of given order N. For each value of N, the Birkhoff normalisations and the parameters of the focus-focus dynamics are represented by polynomials whose coefficients can be computed iteratively with a computer algebra system; no quadratures, such as those needed to compute action-angle variables of resonant normal forms, are needed. We also provide some numerical demonstrations of the method for values of the mass parameter representative of the Sun–Earth and the Sun–Jupiter cases.
小天体与行星的引力近距离相遇可能会使其轨道参数发生重大变化,这可以利用环形受限三体问题进行研究。在本文中,我们以非线性聚焦-聚焦动力学弧线的形式提供了与平面 CRTBP 次级天体快速近距离相遇的参数表示。这一结果是对用 Levi-Civita 正则化表示的问题的哈密顿的 Birkhoff 正则形式进行显著因式分解的结果。对于每个 N 值,伯克霍夫正则表达式和焦点-焦点动力学参数都用多项式表示,其系数可以用计算机代数系统迭代计算;不需要二次方程,如计算共振正则表达式的作用角变量所需的二次方程。我们还提供了该方法在太阳-地球和太阳-木星情况下质量参数值的一些数值演示。
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Pub Date : 2024-09-09DOI: 10.1088/1361-6544/ad700d
Kai Jiang, Tudor S Ratiu and Nguyen Tien Zung
The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples , where X is a dynamical system (vector field) and is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.
本文的目的是首次提出一种对偶同时局部归一化的一般理论,其中 X 是一个动力学系统(向量场),X 是一个由 X 保留的底层几何结构,即使两者都有奇点。这种耦合自然出现在许多问题中,例如汉密尔顿动力学,其中 X 是交映结构,我们有伯克霍夫正形式理论;或约束动力学,其中 X 是切分子空间的平滑分布,一般是奇异分布,等等。在本文中,几何结构有以下几种类型:体积形式、交映形式、接触形式、泊松张量以及它们的奇异版本。本文主要讨论了当 X 和都是奇异点时较为困难的情况,其结果证明了在这些情况下存在自然的同时正则表达式。一般来说,正化只是形式上的,但当和 X 是(实或复)解析的,并且 X 是解析或达布可积分的,那么同时正化也是解析的。我们的理论基于一种新方法,即 "环守恒原理",以及经典的分步归一化技术和等变路径法。
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