Free time minimizers for the three-body problem
Free time minimizers for the three-body problem
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三体问题的空闲时间最小化
DOI：
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发表时间：
2017
影响因子：
1.6
通讯作者：
Héctor Sánchez Morgado
中科院分区：
文献类型：
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作者：
R. Moeckel;R. Montgomery;Héctor Sánchez Morgado
作用的自由时间极小化（Mañe在蒙得维的亚国际动力系统大会（向Ricardo Mañé致敬），第362卷，第120-131页，1996年）中称为“半静态”解）在汉密尔顿-雅可比方程的弱KAM解理论中发挥着核心作用（Fathi在拉格朗日动力学中的弱KAM定理初步版本10，2017年）。我们证明了牛顿三体问题的任何解（其渐近于拉格朗日的抛物位似解）最终都是自由时间极小化器。相反，我们证明，每个自由时间极小往往拉格朗日的解决方案，提供的质量比在于在一定的大开集的质量比。我们的灵感来自工作的大Luz和Maderna（数学Proc Camb Philos Soc 156：209-227，1980），其中表明，每一个自由时间极小的N体问题是抛物线，因此必须渐近的一套中心配置。我们排除了由第二个变分参数渐近欧拉的中心配置。中心构型对应于麦基希爆炸动力学的静止点。质量比的大开集是指在每个欧拉静止点处的线性化动力学具有复特征值的质量比。
Free time minimizers of the action (called “semi-static” solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120–131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton–Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton’s three-body problem which is asymptotic to Lagrange’s parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange’s solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209–227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler’s central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.