Nonholonomic Controlled Hamiltonian System: Symmetric Reduction and Hamilton-Jacobi Equations

In the paper’s Type I Hamilton–Jacobi theorem for distributional RCH systems, the author claims that the sole hypothesis that γ is closed on D (together with Im γ ⊂ M and Im Tγ ⊂ K) implies Tγ·X̃γ = XK·γ. The proof relies on equation (3.1) and a nondegeneracy argument but implicitly assumes the desired equality to transform the right-hand side into τK iM* dγ(...)=0; the concluding step is circular and does not establish the implication from closedness alone. In contrast, the model shows that the defect δ := XK∘γ − Tγ·X̃γ satisfies ωK(δ, ·) = d(H|M ∘ γ)(TπQ·), so δ = 0 if and only if d(H|M ∘ γ) annihilates D — the standard nonholonomic HJ PDE. Thus the paper’s Type I statements (and their reduced variants) are missing a necessary hypothesis. The paper’s Type II theorem is correct and aligns with the model’s argument using naturality of Hamiltonian vector fields and τK·XH = XK on M.