Isodynamicity as a Dirac Constraint on the Ideal-MHD Lie–Poisson Bracket: From the Axisymmetric X-Point to the Homoclinic Tangle

AbstractMotivated by correspondence with A. Gruber and Sina (Sandia National Laboratories) on ex-tending Lie–Poisson bracket reductions to semidirect-product groups, and with D. Peralta-Salas(ICMAT) on the geometry of isodynamic equilibria, this Jalon note examines whether the isody-namic constraint |B|2 = B0 (ψ)2 on a magnetic flux surface can be treated as a genuine second-class constraint, à la Dirac, on the noncanonical (Morrison–Greene–Newcomb) Poisson bracketof ideal MHD, rather than as a soft penalty term in a stellarator-optimization cost functional.Building on the precedent of Chandre, Morrison & Tassi (2012) for the incompressibility con-straint, we derive the secondary constraint Φ2 = {Φ1 , H} explicitly, verify the underlying vectoridentity symbolically, and compute the leading (principal-symbol) structure of the Dirac matrixC12 (x, y) = {Φ1 (x), Φ2 (y)}. We show that C12 is generically elliptic away from the gauge-typedegeneracy k ∥ B (field-line relabeling), and that — contrary to a heuristic conjecture advancedby the author in preliminary correspondence — the axisymmetric magnetic X-point is not alocus of enhanced degeneracy: it is, in this reduced two-dimensional setting, the point of max-imal ellipticity, since the toroidal field there is exactly orthogonal to every admissible poloidalwavevector. We further test, in a Chirikov two-resonance island-overlap toy model, whetherany dynamically relevant direction — the resonant wavevector, the generic chaotic (Lyapunov)stretching direction, or the actual unstable manifold of the homoclinic tangle — comes closeto the Dirac degeneracy direction k ∥ B. All three land within 10◦ of k ⊥ B, the diametricalopposite, reflecting a structural anisotropy of magnetized plasma dynamics rather than a searchfailure. We present this as a corrected, well-triangulated pessimistic Jalon result, and outlinewhat a genuinely three-dimensional (non-axisymmetric-equilibrium) extension would need toshow to overturn it.AI-contribution disclosure. This note was produced in collaborative dialogue with an AIlanguage model (Anthropic Claude), which assisted with literature retrieval, symbolic verification(via sympy) of the vector identities and tensor contractions in Sec. 3, the numerical toy-modelintegration, Newton-method fixed-point search, and manifold tracing in Sec. 5, and drafting. Allphysical framing, the decision to pursue and then correct the X-point conjecture, the choice toescalate to the two-resonance triangulation, and responsibility for the final content rest with thehuman author. No institutional funding or affiliation supports this work.