EXISTENCE OF INVARIANT VOLUMES IN NONHOLONOMIC SYSTEMS

The paper’s main theorem states that a natural nonholonomic system admits a base-dependent invariant volume iff there exists ρ in Γ(D^0) with ϑ_C + ρ exact, where ϑ_C = m^{αβ} L_{W_α}η_β; in that case exp(π_Q^* g)·μ_C is preserved when ϑ_C + ρ = dg. This is explicitly asserted as Theorem 1.1 and Theorem 5.3, with the Leray-type definition of μ_C and a global divergence computation on T*Q (via Ξ_H^C) yielding div_{μ_C} X = −n·π_Q^*⟨ϑ_C, v⟩ on D* . The candidate solution reproduces the same criterion and invariant form using a different route (local Leray contraction and μ-derivatives of multipliers), and matches the paper’s statement about exp(π_Q^* g)·μ_C, but it omits the global factor n in the divergence identity. The paper’s own derivation clearly carries an n (cf. eqs. (6),(8) and the proof of Theorem 5.3) , while its summary statement about the preserved density writes exp(π_Q^* g) rather than exp(π_Q^* n g) . Since the existence criterion is unaffected by a global constant factor (n can be absorbed into g or into the definition of ϑ_C), both the paper and the model are correct on the equivalence and preserved measure claim; the model’s proof is different in method and, like the paper’s summary line, suppresses the factor n that appears in the detailed divergence computation.