Kozai–Lidov cycles = simple pendulum

The quadrupole Kozai mechanism, which describes the hierarchical three-body problem for a test particle when the gravitational potential of the tertiary is expanded to leading order (quadrupole) in the ratio of semimajor axes, a/a_per ≪ 1, and is doubly averaged over the orbits, is shown to be equivalent to a simple pendulum. The change in the eccentricity squared equals the height of the pendulum from its lowest point: e_max² − e² = h = l (1 − cos θ). In particular, this results in useful expressions for the Kozai-Lidov cycles (KLC) period, and the maximal and minimal eccentricities in terms of orbital constants. We derive the equivalence using the vector coordinates α = j + e, β = 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 α and β simplify to α = 2∂_αφ × α and β = 2∂_βφ × β, where φ is the normalized averaged interaction potential, and are symmetric under α ↔ β for the KLC quadratic potential. Their constraints simplify to α² = β² = 1, and they are distributed uniformly and independently on the unit sphere for a uniform distribution in phase space (with a fixed energy).