On the dynamics of a heavy symmetric ball that rolls without sliding on a uniformly rotating surface of revolution

Proposition 8 of the paper defines Λ = 2 p1 F(p1)^2 ∂_{p2} ∧ (∂_{p1} + (G3 p4 + Ω g3)∂_{p3} + (G4 p3 + Ω g4)∂_{p4}) on M◦4 and asserts: (i) X◦ = Λ(dE◦,·); (ii) Λ is a rank‑two Poisson tensor; (iii) the components of J◦ are Casimirs. The authors prove (i) by writing Λ = (2 p1/p2) F^2 ∂_{p2} ∧ X◦ on {p2 ≠ 0} and using L_{X◦}E◦ = 0, then extend by continuity; they justify (ii) noting the characteristic distribution is spanned by ∂_{p2} and ∂_{p1} + (G3 p4 + Ω g3)∂_{p3} + (G4 p3 + Ω g4)∂_{p4}; and they verify (iii) via U′=GU and u′=Gu+g (Proposition 8 and surrounding formulas E◦ (19), J (20), and X (9)–(11) in the PDF). The candidate solution proves the same statements but with a different route: it decomposes Λ = ρ Y ∧ V (ρ=2 p1 F^2, Y=∂_{p2}, V=∂_{p1}+α∂_{p3}+β∂_{p4}), shows [Y,V]=0 (since α,β are p2‑independent) and applies the Schouten bracket identity to conclude [Λ,Λ]=0; for (i) it computes Λ(dE◦,·) component‑wise and matches X◦, with the final ∂_{p2}‑component either by direct differentiation or by the same continuity trick as the paper; and for (iii) it shows V(J◦)=0 using U′ and u′, hence Λ(dJ◦,·)=0. Thus the content agrees with the paper, but the proof details are different. Minor issues: the model’s outline briefly (and unnecessarily) claims V’s coefficients depend only on p1 (they also depend on p3,p4), but its detailed proof only uses p2‑independence; the paper’s proof of the Jacobi identity is terse, whereas the model supplies the missing Schouten‑bracket calculation. Citations: Proposition 8 and its proof sketch, including the form of Λ and the spanning fields, appear on p.16 of the PDF ; the explicit formulas for X (9)–(11) used to match components are on p.9–10 ; E◦ (19) and J (20) are on p.10 ; the paper’s line Λ = (2 p1/p2)F^2∂_{p2}∧X◦ and the continuity argument for (i) are also on p.16 .