Nonplanar Periodic Solutions for Spatial Restricted 3-Body and 4-Body Problems

2012/09/14 by Xiaoxiao Zhao, Zhao, Xiaoxiao, Shiqing Zhang +1
Engineering · #FOS: Physical sciences #Mathematical Physics (math-ph) #Spacecraft Dynamics and Control

paper · pdf · doi:10.48550/arxiv.1209.3254

openalex publication_date 2012/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the existence of non-planar periodic solutions for the following spatial restricted 3-body and 4-body problems: for N=2 or 3, given any masses m₁,...,m_N, the mass points of m₁,...,m_N move on the N circular obits centered at the center of masses, the sufficiently small mass moves on the perpendicular axis passing the center of masses. Using variational minimizing methods, we establish the existence of the minimizers of the Lagrangian action on anti-T/2 or odd symmetric loop spaces. Moreover, we prove these minimizers are non-planar periodic solutions by using the Jacobi's Necessary Condition for local minimizers.

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