Kozai–Lidov cycles = simple pendulum
Roi D. Basha, Ygal Klein, Boaz Katz · Monthly Notices of the Royal Astronomical Society Letters · 2025
ABSTRACT 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, $a/a_{\text{per}}\ll 1$, 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: $e_{\text{max}}^2-e^2=h=l\left(1-\cos {\theta }\right)$. 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 $\boldsymbol{\alpha }={\bf \boldsymbol{ j}}+{\bf \boldsymbol{ e}}, \boldsymbol{\beta }={\bf \boldsymbol{ j}}-{\bf \boldsymbol{ e}}$ for the inner Keplerian orbit, where ${\bf \boldsymbol{ j}}$ is the normalized specific angular momentum, and ${\bf \boldsymbol{ e}}$ is the eccentricity vector. The equations of motion for $\boldsymbol{\alpha }$ and $\boldsymbol{\beta }$ simplify to $\dot{\boldsymbol{\alpha }}=2\partial _{\boldsymbol{\alpha }} \phi \times \boldsymbol{\alpha }$ and $\dot{\boldsymbol{\beta }}=2\partial _{\boldsymbol{\beta }} \phi \times \boldsymbol{\beta }$, where $\phi$ is the normalized averaged interaction potential, and are symmetric under $\boldsymbol{\alpha } \leftrightarrow \boldsymbol{\beta }$ for the KLC quadratic potential. Their constraints simplify to $\boldsymbol{\alpha }^2=\boldsymbol{\beta }^2=1$, and they are distributed uniformly and independently on the unit sphere for a uniform distribution in phase space (with a fixed energy).