A family of maps and a vector field on plane polygons

The paper explicitly states and proves (for triangles) that the vector field ξ on the translation-quotient T is Liouville integrable with Poisson-commuting integrals perimeter P and area A, and that the discrete relation t∼ preserves area (Theorem 1), deriving the Hamiltonian structure from a symplectic 2-form ω and showing dA = i_η ω where η is the rotation field; since P is rotation-invariant, {P, A} = 0. These ingredients appear in Lemmas 3.2–3.4 and Section 4 of the paper, with Theorem 1 summarizing the triangle case. The candidate solution follows essentially the same structure: (i) identifies ω; (ii) shows i_ξ ω = dP and computes the Hamiltonian vector field of area as a rotation; (iii) uses rotation-invariance of P to get {P, A} = 0; (iv) verifies generic independence of dP and dA; and (v) proves area invariance for t∼ by a bilinear determinant expansion with telescoping cancellations. This matches the paper’s core approach and conclusions, with only minor normalizations and sign conventions differing and with the candidate providing an explicit independence check that the paper leaves implicit. See Theorem 1 and its surrounding discussion (triangles) and Lemmas 3.2–3.5 (odd-gons and the symplectic/Hamiltonian framework) in the paper . The definition of ξ and the odd-n symplectic form ω are stated in Section 2–3 . The area-preservation for t∼ is established generally (as invariance of the algebraic multi-area), which for n = 3 reduces to Euclidean area invariance .