Exact closed-form solutions to hierarchical many-body problems via fractional dynamics and elementary function representations

We present exact closed-form analytical solutions to the radial infall problem in hierarchical many-body configurations governed by fractional dynamics. Beginning from power-law correlation functions characteristic of hierarchical gravitational systems with inner-subsystem permutation symmetry, we rigorously derive the scaling law αk=2−2/(Nk+1) connecting fractional parameters to subsystem particle number. For the hierarchical three-body configuration with α=3/2, the effective radial equation of motion is r̈3=−ωeff2r3−1/2, admitting exact solutions in two equivalent forms: a parametric trigonometric representation r3(θ)=a4sin4θ and a direct temporal inversion via Chebyshev polynomials, both valid over the full physical domain. These solutions conserve energy to machine precision (∼10−16) and achieve 20–334× computational speedup over adaptive numerical integration. We further establish that the near-collision approach exponent is γ=1 universally for all Nk≥1, with αk governing the rate of the leading correction; this dual role of αk provides a natural physical interpretation of the scaling law as a measure of how singular the approach to collision is at each hierarchical level. Systematic generalization to N-body hierarchies via degree-(2Nk−1) polynomial equations establishes a complete analytical framework, reducing computational complexity from O(N3) to O(NlogN) while maintaining analytical exactness. These results apply specifically to hierarchical configurations satisfying the separation condition r3≫a12 with radial infall initial conditions and do not supersede Poincaré’s theorem for the general three-body problem.