Periodic perturbations of central force problems and an application to a restricted 3-body problem

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Alberto Boscaggin , Walter Dambrosio , Guglielmo Feltrin
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引用次数: 0

Abstract

We consider a perturbation of a central force problem of the formx¨=V(|x|)x|x|+εxU(t,x),xR2{0}, where εR is a small parameter, V:(0,+)R and U:R×(R2{0})R are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (ε=0) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at ε=0. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential V(r)=κ/rα for α(,2){2,0,1}). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction.

中心力问题的周期性扰动及其在受限三体问题中的应用
我们考虑形式为x¨=V′(|x|)x|x|+ε∇xU(t,x),x∈R2∖{0}的中心力问题的扰动,其中ε∈R是一个小参数,V:(0,+∞)→R和U:R×(R2∖{0})→R是光滑函数,U是第一变量中的τ周期。基于为无扰动问题(ε=0)引入合适的时间映射(径向周期和梢角)以及相关的非退化条件,我们应用高维版本的 Poincaré-Birkhoff 定点定理证明了从ε=0 处的不变环分岔出的非圆形 τ 周期解的存在性。然后,我们证明在一些具体的物理实例中(包括α∈(-∞,2)∖{-2,0,1}的均相势能 V(r)=κ/rα ),这个非退化条件是满足的。最后,还给出了非牛顿相互作用的受限三体问题的应用。
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
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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