Variational Persistence of Periodic and Quasi-Periodic Orbits in Symmetric N-Body Problems
Abstract
Many mechanical systems, such as the Newtonian n-body system, are invariant under rotations. The rotational symmetry of the system gives rise to two commuting actions: the dynamical action and the action of the symmetry group. In this paper, we exploit this symmetry to construct families of periodic and quasi-periodic trajectories. In the strong-force setting, we extend Montgomery's variational approach to free homotopy classes of curves that are periodic up to a prescribed nontrivial rotation. In the weak-force setting, we establish, under suitable hypotheses, global and local variational persistence results for action minimizers in loop spaces defined by general finite symmetry-group actions under sufficiently small rotations. As applications, we apply these results to the figure-eight orbit, under a numerically supported assumption of strict local minimality, to double choreographic loops with q-fold rotation symmetry, and to counter-rotating double choreographic loops in planar (N+3)-body problems.
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BibTeX
@misc{liang2026variational,
title = {{Variational Persistence of Periodic and Quasi-Periodic Orbits in Symmetric $N$-Body Problems}},
author = {Hangqi Liang and Zhihong Xia and Tingjie Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10101},
eprint = {2610.10101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10101},
}
RIS
TY - UNPB
TI - Variational Persistence of Periodic and Quasi-Periodic Orbits in Symmetric N-Body Problems
AU - Hangqi Liang
AU - Zhihong Xia
AU - Tingjie Zhou
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10101
UR - https://arxiv.org/abs/2610.10101
AB - Many mechanical systems, such as the Newtonian n-body system, are invariant under rotations. The rotational symmetry of the system gives rise to two commuting actions: the dynamical action and the action of the symmetry group. In this paper, we exploit this symmetry to construct families of periodic and quasi-periodic trajectories. In the strong-force setting, we extend Montgomery's variational approach to free homotopy classes of curves that are periodic up to a prescribed nontrivial rotation. In the weak-force setting, we establish, under suitable hypotheses, global and local variational persistence results for action minimizers in loop spaces defined by general finite symmetry-group actions under sufficiently small rotations. As applications, we apply these results to the figure-eight orbit, under a numerically supported assumption of strict local minimality, to double choreographic loops with q-fold rotation symmetry, and to counter-rotating double choreographic loops in planar (N+3)-body problems.
ER -