Periodic Oscillations in a 2N-Body Problem

The paper proves the existence of a local family of double-symmetric periodic hip-hop solutions via an implicit-function argument around the circular base state, using the reduced system r¨=a^2/r^3−2rm f(r,d), d¨=−(m/2) d g(r,d) with f,g as in (1)–(3), and the definitions of αN,γN in (5). It shows D(a,b,t)=b D̂(a,b,t), computes D̂t at b=0 to get T(0)=π/2 sqrt(r0^3/(m αN)), and establishes the non-degeneracy needed for IFT by analyzing ∂a Rt and ∂T D̂t at the base point, yielding a 4T(b)-periodic branch with a(0)=√(m γN r0) and T(0)=π/2 sqrt(r0^3/(m αN) (Theorem 3) . The candidate solution reproduces the IFT scheme (R_t and D scaled by b) and the symmetry gluing, but it silently drops the factor 1/2 in the vertical equation, uses d¨=−m d g, and therefore derives the wrong base quarter-period T0=(π/2) sqrt(r0^3/(2 m αN)). It then attributes the discrepancy to a “harmless normalization,” which is incorrect—the difference comes from the equation itself, not the definition of αN. Aside from this constant error, the method (D=b D̂, Jacobian triangularity, time-reversal and d↦−d symmetries) matches the paper’s strategy. Two small notes on the paper: (i) the text briefly invokes Proposition 2 to conclude ωr T0≠pπ; as stated (γN/αN≤4+√2/8), this bound alone does not preclude the case p=1 exactly, so one should exclude equality by a sharper estimate or an explicit check, and (ii) there is a harmless typo where Dbtt is evaluated at T0/2 when the cross-product is at T0. These do not affect the result’s correctness. Overall, the paper’s argument is correct; the model’s solution is incorrect on the governing equation and the resulting value of T(0) .