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.In...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.展开更多
In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints.Our approach is based on some novel features of the Keplerian action functional...In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints.Our approach is based on some novel features of the Keplerian action functional,constraint convex optimization techniques,and variational methods.We demonstrate the strength of this method by constructing relative periodic solutions for the planar four-body problem within a special topological class,and our results hold for an open set of masses.展开更多
文摘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.
文摘In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints.Our approach is based on some novel features of the Keplerian action functional,constraint convex optimization techniques,and variational methods.We demonstrate the strength of this method by constructing relative periodic solutions for the planar four-body problem within a special topological class,and our results hold for an open set of masses.
