{"title":"Three-Body 3D-Kepler Electromagnetic Problem—Existence of Periodic Solutions","authors":"V. Angelov","doi":"10.3390/appliedmath4020034","DOIUrl":null,"url":null,"abstract":"The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom.","PeriodicalId":503400,"journal":{"name":"AppliedMath","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"AppliedMath","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/appliedmath4020034","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom.
本文的主要目的是证明三维开普勒公式中三体问题周期解的存在性。我们已经解决了在外部惯性系中考虑三个粒子时的相同问题。我们从三体运动方程入手,三体运动方程是(我们之前推导的)任意数目物体运动方程的子集。在闵科夫斯基空间,有 12 个运动方程。事实证明,其中三个是其他九个的结果,因此它们的数量变成了九个,就像未知轨迹一样多。开普勒公式假定一个粒子(原子核)位于坐标原点。其他两个粒子的运动由一个中性系统来描述,与未知速度有关。由于有限的真空光速,产生了与状态相关的延迟。我们得到了球面坐标下的运动方程,并将其分为两组。在第一组中,所有未知函数的参数都是延迟。我们将它们的解作为初始函数。然后,其余两个粒子的运动方程必须在初始点右侧求解。为了证明周期解的存在性和唯一性,我们选择一个由满足某些补充条件的周期性无限平稳函数组成的空间。然后,我们使用一个合适的算子,该算子作用于这些空间,其固定点为周期解。我们应用作用于周期函数空间的算子的定点定理。通过这种方法,我们证明了 He 原子在经典电动力学框架下的稳定性。在我们之前的一篇论文中,我们证明了平面运动自旋函数的存在。因此,我们证实了玻尔和萨默菲尔德关于氦原子的假设。