Affine generalizations of the nonholonomic problem of a convex body rolling without slipping on the plane

The paper’s Proposition 5.1 asserts that, for the homogeneous sphere system (5.1)–(5.2), if div_R^2 Vs ≡ 0 and div_S^2 W ≡ 0 then the product measure dM dudαdβdγ is invariant, and sketches that the proof is a direct computation using the intrinsic divergence on S^2 (, ). The candidate solution supplies a detailed divergence computation on the product manifold R^3_M × R^2_u × SO(3)_B (using Haar on SO(3)), splitting the divergence into M-, u-, and SO(3)-blocks, and shows the total divergence is (I/(I+mr^2)) div_R^2 Vs + (mr/(I+mr^2)) div_S^2 W, hence zero under the paper’s hypotheses. This matches the paper’s claim and fills in the omitted steps; the approaches are equivalent in spirit but the candidate provides the full calculation. Key identities and model equations (5.1)–(5.2) and Poisson-vector conventions are consistent with the paper (, ).