On the existence of minimal expansive solutions to the $N$ -body problem

IF 2.6 1区 数学 Q1 MATHEMATICS
Davide Polimeni, Susanna Terracini
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引用次数: 0

Abstract

We deal, for the classical \(N\)-body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unified manner. Our approach is based on the minimization of a renormalized Lagrangian action on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions.

Abstract Image

论N$体问题的最小扩展解的存在性
对于经典的(N)体问题,我们讨论了在规定的渐近方向和体的初始配置下,是否存在作用最小化的半全展开解。我们以统一的方式处理了双曲、双曲-抛物和抛物弧的情况。我们的方法基于对合适函数空间的重规范化拉格朗日作用的最小化。利用这种新策略,我们能够证实双曲和抛物线解存在的已知结果,并首次证明了在一个合适的类中,任何规定渐近展开的双曲抛物线解的存在性。与该类中的每个元素相关联,我们找到了汉密尔顿-雅可比方程的粘性解,作为值函数的线性修正。此外,我们还设法精确描述了抛物线和双曲抛物线解的增长。
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来源期刊
Inventiones mathematicae
Inventiones mathematicae 数学-数学
CiteScore
5.60
自引率
3.20%
发文量
76
审稿时长
12 months
期刊介绍: This journal is published at frequent intervals to bring out new contributions to mathematics. It is a policy of the journal to publish papers within four months of acceptance. Once a paper is accepted it goes immediately into production and no changes can be made by the author(s).
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