Billiard with Coulomb Potential Inside an Elliptic Ring

IF 1.9 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL

Russian Journal of Mathematical Physics Pub Date : 2026-07-15 DOI:10.1134/S1061920826600315

G.V. Belozerov, S.A. Galkin

引用次数: 0

Abstract

We consider a billiard system inside a ring formed by two ellipses with common foci \(F_1\) and \(F_2\) affected by Coulomb forces, condensed in \(F_1\) and \(F_2\) with charges \(\gamma_1\) and \(\gamma_2\), respectively. The reflection from the boundary of the domain is supposed to be essentially elastic. This billiard is Liouville integrable in terms of piecewise smoothness. An explicit formula of the additional first integral was found. We study the topology of the Liouville foliation of this system in several cases: \(\gamma_1<0,\gamma_2=0\) (Kepler case); \(\gamma_1>0,\gamma_2=0\); \(\gamma_1=\gamma_2\). For each of these cases, Fomenko and Fomenko–Zieschang invariants are calculated.

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椭圆环内具有库仑势的台球

我们考虑一个环内的台球系统，该系统由两个具有共同焦点\(F_1\)和\(F_2\)的椭圆组成，受库仑力的影响，分别凝聚在\(F_1\)和\(F_2\)中，电荷分别为\(\gamma_1\)和\(\gamma_2\)。从域的边界反射被认为本质上是弹性的。这个台球在分段平滑性方面是刘维尔可积的。得到了附加第一个积分的显式公式。我们在几种情况下研究了该系统的Liouville叶理的拓扑结构：\(\gamma_1<0,\gamma_2=0\) （Kepler情况）；\(\gamma_1>0,\gamma_2=0\)；\(\gamma_1=\gamma_2\)。对于每种情况，计算Fomenko和Fomenko - zieschang不变量。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Russian Journal of Mathematical Physics 物理-物理：数学物理

CiteScore

3.10

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.