Near tangent dynamics in a class of Hamiltonian impact systems

The candidate solution reproduces the paper’s main local normal form near a grazing tangency and follows essentially the same proof strategy. Specifically: (i) it identifies and proves smoothness of the singularity curve I_ε^{tan}(θ) and uses a symplectic rectification via the generating function S(θ,K)=Kθ+∫ I_ε^{tan}(θ′)dθ′ to set K=I−I_ε^{tan}(θ), φ=θ, exactly as in the paper’s Eq. (70) leading to the form (71a–b) and thence to the normal form (9) . (ii) It derives the square-root singularity in the angle increment from the reduction in travel time near the wall, matching the coefficient −(2ω2(Itan))^{3/2}/|V′1(qw1)| stated in Eq. (10) (the square-root arising from Eq. (7)’s travel-time integral) . (iii) It obtains the first-order action change as a Melnikov-type integral along the ε=0 tangent trajectory, coinciding with the paper’s definition of f in (9) and its derivation in (40–41) . (iv) It states the same remainder structure, including the C^{r−2} loss tied to the √(−K) behavior on the impacting side, as codified in (9)–(10) and in the two-branch precursor formulas (34a–b) . Minor stylistic differences (e.g., the model’s use of a “clearance functional” versus the paper’s star-section construction and forward/backward flow to/ from the wall) do not affect the result; the paper’s route is more systematic for ensuring smoothness and uniqueness of the defining objects on the section .