ON ACTION-MINIMIZING SOLUTIONS OF THE TWO-CENTER PROBLEM

doi:10.1007/s10473-022-0615-9

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (6): 2450-2458.doi: 10.1007/s10473-022-0615-9

ON ACTION-MINIMIZING SOLUTIONS OF THE TWO-CENTER PROBLEM

Kuo-Chang CHEN

Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, China

收稿日期:2022-07-26 出版日期:2022-12-25 发布日期:2022-12-16
基金资助:
This work was supported in parts by the Ministry of Science and Technology in Taiwan.

ON ACTION-MINIMIZING SOLUTIONS OF THE TWO-CENTER PROBLEM

Kuo-Chang CHEN

Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, China

Received:2022-07-26 Online:2022-12-25 Published:2022-12-16
Supported by:
This work was supported in parts by the Ministry of Science and Technology in Taiwan.

摘要/Abstract

摘要： The two-center problem, also known as Euler’s three-body problem, is a classic example of integrable systems. Among its periodic solutions, planetary type solutions are periodic solutions which enclose both centers. Inspired by advances on n-body and n-center problems via variational techniques developed during the past two decades, a recent paper (Arch. Rat. Mech. Ana. 2022) shows the minimizing property of planetary type solutions for any given masses of centers at fixed positions, as long as the period is above a mass-dependent threshold value. In this paper, we provide further discussions regarding this minimizing approach. In particular, we improve the above-mentioned mass-dependent threshold value by refining estimates for action values.

关键词: wo-center problem, n-body problem, variational method, collision singularity

Abstract: The two-center problem, also known as Euler’s three-body problem, is a classic example of integrable systems. Among its periodic solutions, planetary type solutions are periodic solutions which enclose both centers. Inspired by advances on n-body and n-center problems via variational techniques developed during the past two decades, a recent paper (Arch. Rat. Mech. Ana. 2022) shows the minimizing property of planetary type solutions for any given masses of centers at fixed positions, as long as the period is above a mass-dependent threshold value. In this paper, we provide further discussions regarding this minimizing approach. In particular, we improve the above-mentioned mass-dependent threshold value by refining estimates for action values.

Key words: wo-center problem, n-body problem, variational method, collision singularity

中图分类号:

70F05

Kuo-Chang CHEN. ON ACTION-MINIMIZING SOLUTIONS OF THE TWO-CENTER PROBLEM[J]. 数学物理学报(英文版), 2022, 42(6): 2450-2458.

Kuo-Chang CHEN. ON ACTION-MINIMIZING SOLUTIONS OF THE TWO-CENTER PROBLEM[J]. Acta mathematica scientia,Series B, 2022, 42(6): 2450-2458.

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ON ACTION-MINIMIZING SOLUTIONS OF THE TWO-CENTER PROBLEM

ON ACTION-MINIMIZING SOLUTIONS OF THE TWO-CENTER PROBLEM

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