THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS

The paper’s Proposition 10 states exactly the Dirichlet boundary characterization of R-symmetric spin–spin resonant orbits and the four-type classification in the balanced case, using the reversing symmetry R_{α,β1,β2}, the fixed-point criterion, and periodicities; its proof reduces item (i) to the spin–orbit Proposition 5 and gives the balanced-case details (items (i)–(ii)) . The candidate solution reproduces the same structure: the same reversing symmetry, the “orbit meets Fix(R)” criterion, the symmetry identity θ_j(t)=2β_j−θ_j(2α−t), the boundary conditions θ_j(α)=β_j and θ_j(α+nπ)=β_j+m_jπ, and the four independent balanced types with (β_1,β_2)∈{0,π/2}^2; it also supplies a constructive converse via reversibility and π-periodicity of W, which is fully consistent with the paper’s argument (and slightly more explicit) . The only minor imprecision in the model write-up is the statement that a time shift by π identifies α=0 and α=π; the system is 2π-periodic in time, and the equivalence of representative types in the balanced case is established in the paper by parity/normalization of boundary conditions rather than an invariance under t↦t+π .