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arXiv · 2610.25355

A coupling normal form for reversible center-center Hamiltonians with application to the restricted three-body problem

Mingpei Lin
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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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Mingpei Lin. 2026-10-20. A coupling normal form for reversible center-center Hamiltonians with application to the restricted three-body problem. https://arxiv.org/abs/2610.25355

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