Hamilton-Jacobi Equations for Nonholonomic Magnetic Hamiltonian Systems

Theorem 5.3 of Wang (2021) explicitly proves the equivalence between the two Type II Hamilton–Jacobi conditions on the reduced nonholonomic magnetic system by pairing with the reduced distributional two-form and invoking non-degeneracy; see the statement of the theorem and the key equality ω^B_{K̄}(T λ̄ · X^B_H · ε − τ_{K̄} · T ε̄ · X^B_{h_{K̄}·ε̄}, T λ̄ · w) = 0 leading to the equivalence T γ̄ · X_ε = X^B_{K̄} · ε̄ ⇔ T λ̄ · X^B_H · ε = τ_{K̄} · T ε̄ · X^B_{h_{K̄}·ε̄} (non-degeneracy then gives equality of vectors) . The proof uses the reduction identities π*/G ω^B_{K̄} = ω^B_U and the defining reduced equation i_{X^B_{K̄}} ω^B_{K̄} = d h_{K̄} . By contrast, the candidate solution tacitly assumes extra hypotheses (e.g., Im(Tγ) ⊂ U and ε(M) ⊂ M), omits the necessary projections τ_U/τ_{K̄} at intermediate steps, and pairs at points where K̄ may be undefined, so it does not form a correct proof as written.