Oscillating about coplanarity in the 4 body problem

• 成果类型: Article
• 署名作者: Montgomery, Richard
• 署名单位: University of California System; University of California Santa Cruz
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00879-0
• 发表日期: 2019
• 页码: 113-144
• 关键词: hamilton-jacobi equations; 3-body problem

摘要

For the Newtonian 4-body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants, that is, times at which all 4 bodies lie in the same plane. This result generalizes a known result for collinear instants (syzygies) in the zero angular momentum planar 3-body problem, and extends to thed + 1 body problem in d-space. The proof begins by identifying the translation-reduced configuration space with real dxd matrices, the degeneration locus (set of coplanar configurations when d = 3) with the set of matrices having determinant zero, and the mass metric with the Frobenius (standard Euclidean) norm. Let S denote the signed distance from a matrix to the hypersurface of matrices with determinant zero. The proof hinges on establishing a harmonic oscillator type ODE for S along solutions. Bounds on inter-body distances then yield an explicit lower bound. for the frequency of this oscillator, guaranteeing a degeneration within every time interval of length p/.. The non-negativity of the curvature of oriented shape space (the quotient of the translation-reduced configuration space by the rotation group) plays a crucial role in the proof.