{"title":"Shooting for collinear periodic orbits in the Helium model","authors":"Lei Zhao","doi":"10.1007/s00033-023-02120-8","DOIUrl":null,"url":null,"abstract":"Abstract The frozen-planet periodic orbit of the classical collinear Helium model with negative energy is shown to exist by a simple shooting argument. This simplifies the approach established in Cieliebak et al. (Ann Inst H Poincaré Anal Non Linéaire 40:379–455, 2022). With this argument, it also follows that the algebraic count of the number of such orbits with a given negative energy is 1, as recently established in Cieliebak et al. (Nondegeneracy and integral count of frozen-planet orbits in helium, 2022. arXiv:2209.12634 ). The same argument also leads to the existence of other collinear periodic orbits of the classical collinear Helium model.","PeriodicalId":54401,"journal":{"name":"Zeitschrift fur Angewandte Mathematik und Physik","volume":"147 1","pages":"0"},"PeriodicalIF":1.7000,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Zeitschrift fur Angewandte Mathematik und Physik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00033-023-02120-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract The frozen-planet periodic orbit of the classical collinear Helium model with negative energy is shown to exist by a simple shooting argument. This simplifies the approach established in Cieliebak et al. (Ann Inst H Poincaré Anal Non Linéaire 40:379–455, 2022). With this argument, it also follows that the algebraic count of the number of such orbits with a given negative energy is 1, as recently established in Cieliebak et al. (Nondegeneracy and integral count of frozen-planet orbits in helium, 2022. arXiv:2209.12634 ). The same argument also leads to the existence of other collinear periodic orbits of the classical collinear Helium model.
摘要通过一个简单的射击论证，证明了负能量的经典共线氦模型的冰冻行星周期轨道的存在。这简化了Cieliebak等人建立的方法(Ann Inst H poincar Anal Non linsamaire 40:379-455, 2022)。根据这一论点，也可以得出，具有给定负能量的这种轨道的代数计数为1，正如最近在Cieliebak等人(氦中冻结行星轨道的非简并和积分计数，2022)中所建立的那样。arXiv: 2209.12634)。同样的论点也导致了经典共线氦模型的其他共线周期轨道的存在。
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