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 $H(0,q_2,p_1,p_2)=E$, 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 $Σ_E^{\mathrm{sec}}=Σ_E\cap C$ 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.