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arXiv · 2608.12901

Relative Periodic Orbits in the Gutzwiller-type Anisotropic Kepler Problem and $n$-body Problem

Xijun Hu·Yuwei Ou·Zhiwen Qiao·Yifan Yang
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Abstract

We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the $z$-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in \cite{CHHL23}, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the $(1+2n)$-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the $n$-pyramidal problem.

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Xijun Hu, Yuwei Ou, Zhiwen Qiao, Yifan Yang. 2026-08-13. Relative Periodic Orbits in the Gutzwiller-type Anisotropic Kepler Problem and $n$-body Problem. https://arxiv.org/abs/2608.12901

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