A coupling normal form for reversible center-center Hamiltonians with application to the restricted three-body problem
Abstract
Near a reversible center-center equilibrium, the Birkhoff normal form does not describe the families bifurcating from the continued tori, which resonant normal forms capture through a reduced system.With a reflection symmetry, we develop a coupling normal form in which the bifurcating families also appear as tori of action normal forms. Along a selected coupling direction, the first harmonic in the angle difference is absorbed into an action-dependent counterterm. We prove that the coupled recursion determines unique counterterm coefficients, a formal action normal form, and an exact counterterm factorization. The factorization gives a recovery constraint, whose trivial factor corresponds to the Birkhoff normal form. At every finite order, a symmetric pair of bifurcating branches leaves the trivial branch at a critical curve, carrying approximately invariant tori and circles with residual estimates. In the spatial circular restricted three-body problem, these give approximate halo orbits and quasi-halo tori.
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BibTeX
@misc{lin2026coupling,
title = {{A coupling normal form for reversible center-center Hamiltonians with application to the restricted three-body problem}},
author = {Mingpei Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25355},
eprint = {2610.25355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25355},
}
RIS
TY - UNPB
TI - A coupling normal form for reversible center-center Hamiltonians with application to the restricted three-body problem
AU - Mingpei Lin
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.25355
UR - https://arxiv.org/abs/2610.25355
AB - Near a reversible center-center equilibrium, the Birkhoff normal form does not describe the families bifurcating from the continued tori, which resonant normal forms capture through a reduced system.With a reflection symmetry, we develop a coupling normal form in which the bifurcating families also appear as tori of action normal forms. Along a selected coupling direction, the first harmonic in the angle difference is absorbed into an action-dependent counterterm. We prove that the coupled recursion determines unique counterterm coefficients, a formal action normal form, and an exact counterterm factorization. The factorization gives a recovery constraint, whose trivial factor corresponds to the Birkhoff normal form. At every finite order, a symmetric pair of bifurcating branches leaves the trivial branch at a critical curve, carrying approximately invariant tori and circles with residual estimates. In the spatial circular restricted three-body problem, these give approximate halo orbits and quasi-halo tori.
ER -