Local minimality properties of circular motions in 1/rα potentials and of the figure-eight solution of the 3-body problem

The paper proves sufficiency for weak and strong local minimality for periodic problems via a Riccati identity and a Hamilton–Jacobi (HJ) construction: (i) under (L') plus (SR), Q(v) > 0 on periodic variations; (ii) adding (W') and a perturbed Riccati solution W̃ with W̃(T)−W̃(0) ≻ 0 yields an HJ inequality implying strong local minimality. This is stated and derived in Theorem 3.11 and Theorem 4.4, with the HJ verification deferred to Clarke–Zeidan [8] and the existence of W̃ obtained via an embedding theorem (Hestenes) . The candidate solution follows the same structure: it uses the same Riccati completion-of-squares identity to get positivity of Q, builds a perturbed Riccati solution W_ε, and derives an HJ-type pointwise inequality with an explicit feedback v_ε to obtain strong local minimality under (W'). The candidate adds an H^1-coercivity estimate for Q that the paper does not spell out but is consistent with the Riccati identity. One minor difference is that the paper relies on an embedding theorem to obtain W̃ solving Ẇ̃−C+W̃A+A^TW̃+W̃BW̃=−ε0I with W̃(T)−W̃(0)≻0, whereas the candidate appeals to continuity in ε for the Riccati ODE; both routes are standard and compatible. Overall, both are correct and methodologically aligned.