Analysis of point-contact models of the bounce of a hard spinning ball on a compliant frictional surface

The paper proves that the lift‑off–while–rolling point s is a visible two‑fold (σ1=+1, σ2=−1) and derives ν1ν2 = 1 + (20/7)(∂X′υ·X′)/(∂Y′ΛN·Y′) g ε^3 + O(ε^4), which is <1 under their hypotheses, implying a saddle in the reduced sliding dynamics and a codimension‑one rolling lift‑off set; almost all trajectories lift off under slip. These steps and conclusions match the candidate solution’s four-part outline, including the use of H = X′+Ω, the Filippov normal form on z1=0 with matrix [[ν2, −1], [−1, ν1]], the sign reasoning (Y′(s)>0; ∂YΛN<0), and the physical interpretation separating topspin/backspin at lift-off. See the paper’s equations for (17)–(21), (42)–(50), and especially (51)–(54) for the key signs and ν1ν2 formula, and the discussion that only a single trajectory passes through s in the visible case ν1ν2<1 . The restitution-phase requirement Y′(s)>0 is also established earlier for the roll–slip tangencies .