Reconstruction through a coisotropic reduction: the missing momentum on a Poincaré section as a branched covering
Abstract
Determining the momentum missing on a Poincaré section of a two-degree-of-freedom Hamiltonian is usually posed as the root-finding problem H(0,q2,p1,p2)=E, which degenerates at the boundary of the energetically allowed region. We argue that the degeneration is the visible symptom of a geometric object and reorganize the problem around it. The section is a coisotropic hypersurface, the missing momentum is its characteristic direction, and recovering it is a \emph{reconstruction} through the associated coisotropic reduction. Its carrier is not the fibrewise least-energy function, which we show is informationally incomplete, but the correspondence ΣEsec=ΣE∩ C together with the reducing projection πE, a branched reconstruction over the admissible region whose discriminant is identified with the Hill boundary under a base-side regularity condition. We prove, isolating for each conclusion its minimal hypothesis: that πE is proper and a covering away from the discriminant (coercivity); that the fibres have exactly two points over the interior (unimodality); that along the discriminant πE is a fold (fibrewise nondegeneracy and transversality); and that the discriminant coincides with the topological boundary of the admissible region precisely under a regularity of the least-energy function \emph{on the base}, which no fibrewise hypothesis supplies. Strict convexity of H in the momenta is shown to be a sufficient but non-fundamental bundle of these conditions. The boundary is a discriminant, not a Lagrangian caustic. Only afterwards does the squared fibrewise energy residual Fb=(hb−E)2 appear, as the analytic resolution of the fold; inside the admissible region it is a double well with two global minimizers, so branch selection is a separate rule.
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BibTeX
@misc{chanlopez2026reconstruction,
title = {{Reconstruction through a coisotropic reduction: the missing momentum on a Poincaré section as a branched covering}},
author = {E. Chan-López},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16715},
eprint = {2609.16715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16715},
}
RIS
TY - UNPB
TI - Reconstruction through a coisotropic reduction: the missing momentum on a Poincaré section as a branched covering
AU - E. Chan-López
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16715
UR - https://arxiv.org/abs/2609.16715
AB - Determining the momentum missing on a Poincaré section of a two-degree-of-freedom Hamiltonian is usually posed as the root-finding problem H(0,q_2,p_1,p_2)=E, which degenerates at the boundary of the energetically allowed region. We argue that the degeneration is the visible symptom of a geometric object and reorganize the problem around it. The section is a coisotropic hypersurface, the missing momentum is its characteristic direction, and recovering it is a \emph{reconstruction} through the associated coisotropic reduction. Its carrier is not the fibrewise least-energy function, which we show is informationally incomplete, but the correspondence Σ_E^sec=Σ_E∩ C together with the reducing projection π_E, a branched reconstruction over the admissible region whose discriminant is identified with the Hill boundary under a base-side regularity condition. We prove, isolating for each conclusion its minimal hypothesis: that π_E is proper and a covering away from the discriminant (coercivity); that the fibres have exactly two points over the interior (unimodality); that along the discriminant π_E is a fold (fibrewise nondegeneracy and transversality); and that the discriminant coincides with the topological boundary of the admissible region precisely under a regularity of the least-energy function \emph{on the base}, which no fibrewise hypothesis supplies. Strict convexity of H in the momenta is shown to be a sufficient but non-fundamental bundle of these conditions. The boundary is a discriminant, not a Lagrangian caustic. Only afterwards does the squared fibrewise energy residual F_b=(h_b−E)^2 appear, as the analytic resolution of the fold; inside the admissible region it is a double well with two global minimizers, so branch selection is a separate rule.
ER -