Proof of a conjecture by H. Dullin and R. Montgomery

The paper proves that, in the planar Euler (two fixed centers) problem, the periods T+ and T− are piecewise real-analytic along F0-fibers and become monotone on each connected component, with the monotonicity direction depending on which analytic branch (T↓ or T↑) one is on. Concretely, T↓ is increasing in F̂0 and T↑ is decreasing (Theorem 2.1), and the rotation number W = T−/T+ is piecewise monotone with both increasing and decreasing regimes (Theorem 2.2) . By contrast, the candidate solution asserts uniform directions on every component: T+ strictly increasing, T− strictly decreasing, hence W strictly decreasing. This conflicts with the paper’s finer classification (e.g., WS and WP are increasing, while WL is decreasing) . Moreover, the candidate’s sign computation for ∂F0Q− is incorrect with respect to the paper’s period integral (12), leading to the wrong sign for T−′(F0). The paper’s definitions and mapping to Kepler-type integrals, together with the T↓/T↑ splitting, resolve these issues rigorously (equations (12), (14), (16)–(18)) .