2025/03/25 by Roi D. Basha, Ygal Y. Klein, Basha, Roi D. +3 · 1 citation
Computer Science · #Algorithms and Data Compression #Classical Physics (physics.class-ph) #Cryptographic Implementations and Security #Earth and Planetary Astrophysics (astro-ph.EP) #FOS: Physical sciences #Formal Methods in Verification #Solar and Stellar Astrophysics (astro-ph.SR)
paper · pdf · doi:10.48550/arxiv.2503.19972
openalex publication_date 2025/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The quadrupole Kozai mechanism, which describes the hierarchical three-body problem in the leading order, is shown to be equivalent to a simple pendulum where the change in the eccentricity squared equals the height of the pendulum from its lowest point: emax2-e2=h=l(1-cosθ). In particular, this results in useful expressions for the KLC period, and the maximal and minimal eccentricities in terms of orbital constants. We derive the equivalence using the vector coordinates \boldsymbolα=j+e, \boldsymbolβ=j-e for the inner Keplerian orbit, where j is the normalized specific angular momentum, and e is the eccentricity vector. The equations of motion for \boldsymbolα and \boldsymbolβ simplify to \boldsymbolα=2∂\boldsymbolα ϕ× \boldsymbolα and \boldsymbolβ=2∂\boldsymbolβ ϕ× \boldsymbolβ, where ϕ is the normalized averaged interaction potential and are symmetric to replacing \boldsymbolα and \boldsymbolβ for the KLC quadratic potential. Their constraints simplify to \boldsymbolα2=\boldsymbolβ2=1, and they are distributed uniformly and independently on the unit sphere for a uniform distribution in phase space (with a fixed energy).