Abstract

The article presents an analytical review of works devoted to the quaternion regularization of the singularities of differential equations of the perturbed two-body problem generated by gravitational forces, using the four-dimensional Kustaanheimo–Stiefel variables. Most of these works have been published in leading foreign publications. We consider a new method of regularization of these equations proposed by us, based on the use of two-dimensional ideal rectangular Hansen coordinates, two-dimensional Levi-Civita variables, and four-dimensional Euler (Rodrigues–Hamilton) parameters. Previously, it was believed that it was impossible to generalize the famous Levi-Civita regularization of the equations of plane motion to the equations of spatial motion. The regularization proposed by us refutes this point of view and is based on writing the differential equations of the perturbed spatial problem of two bodies in an ideal coordinate system using two-dimensional Levi-Civita variables to describe the motion in this coordinate system (in which the equations of spatial motion take the form of equations of plane motion) and based on the use of the quaternion differential equation of the inertial orientation of the ideal coordinate system in the Euler parameters, which are the osculating elements of the orbit, as well as on the use of Keplerian energy and real time as additional variables, and on the use of the new independent Sundmann variable. Reduced regular equations, in which Levi-Civita variables and Euler parameters are used together, not only have the well-known advantages of equations in Kustaanheimo–Stiefel variables (regularity, linearity in new time for Keplerian motions, proximity to linear equations for perturbed motions), but also their own additional advantages: 1) two-dimensionality, and not four-dimensionality, as in the case of Kustaanheimo–Stiefel, a single-frequency harmonic oscillator describing in new time in Levi-Civita variables the unperturbed elliptic Keplerian motion of the studied (second) body, and 2) slow change in the new time of the Euler parameters, which describe the change in the inertial orientation of the ideal coordinate system, for perturbed motion, which is convenient when using the methods of nonlinear mechanics. This work complements our review paper [1].

中文翻译：

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引力生成的天体动力学模型奇点的四元数正则化（评论）

摘要

本文对致力于使用四维 Kustaan​​heimo-Stiefel 变量对由引力产生的摄动二体问题的微分方程奇点进行四元数正则化的工作进行了分析回顾。这些作品大部分发表在国外主要刊物上。我们考虑了我们提出的这些方程的正则化新方法，该方法基于使用二维理想直角汉森坐标、二维 Levi-Civita 变量和四维 Euler（Rodrigues-Hamilton）参数。此前，人们认为不可能将著名的平面运动方程的列维-奇维塔正则化推广到空间运动方程。我们提出的正则化反驳了这一观点，它的基础是在理想坐标系中写出两个物体的扰动空间问题的微分方程，使用二维Levi-Civita变量来描述该坐标系中的运动（其中空间运动方程采用平面运动方程的形式），并基于欧拉参数中理想坐标系惯性方向的四元数微分方程，欧拉参数是轨道的密切元素，以及关于使用开普勒能量和实时作为附加变量，以及使用新的独立 Sundmann 变量。简化正则方程，其中列维-奇维塔变量和欧拉参数一起使用，不仅具有 Kustaan​​heimo-Stiefel 变量方程的众所周知的优点（开普勒运动的新时间的正则性、线性度、扰动运动的接近线性方程） ），但也有它们自己的额外优点：1）二维，而不是四维，如 Kustaan​​heimo-Stiefel 的情况，单频谐振子在 Levi-Civita 变量的新时间中描述不受扰动的椭圆开普勒运动2）欧拉参数在新时间的缓慢变化，描述了理想坐标系惯性方向的变化，对于摄动运动，这在使用非线性力学方法时很方便。这项工作补充了我们的综述论文 [1]。