{"title":"Poincaré Second Species Solutions of the General Three-Body Problem","authors":"Sergey V. Bolotin","doi":"10.1134/S1560354726050059","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the plane three-body problem with two of the masses much smaller than the third.\nPeriodic solutions with near collisions of the small bodies were named\nby Poincaré <i>second species solutions</i>.\nPoincaré gave an idea of how to prove their existence, but didn’t provide an actual proof.\nComplete proofs appeared much later and only for the restricted three-body problem.\nWe reduce the problem of second species solutions of the general three-body problem\nto the study of dynamics of a billiard type problem and show how to transform\nthe arguments of Poincaré into a rigorous proof.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"31 Editors:","pages":"738 - 750"},"PeriodicalIF":0.8000,"publicationDate":"2026-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354726050059","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the plane three-body problem with two of the masses much smaller than the third.
Periodic solutions with near collisions of the small bodies were named
by Poincaré second species solutions.
Poincaré gave an idea of how to prove their existence, but didn’t provide an actual proof.
Complete proofs appeared much later and only for the restricted three-body problem.
We reduce the problem of second species solutions of the general three-body problem
to the study of dynamics of a billiard type problem and show how to transform
the arguments of Poincaré into a rigorous proof.
期刊介绍：
Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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