Kozai Lidov Cycles = Simple Pendulum

Abstract

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: $e_{\text{max}}^2-e^2=h=l\left(1-\cos{\theta}\right)$. 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 $\boldsymbol{\alpha}=\textbf{j}+\textbf{e}, \boldsymbol{\beta}=\textbf{j}-\textbf{e}$ for the inner Keplerian orbit, where $\textbf{j}$ is the normalized specific angular momentum, and $\textbf{e}$ 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 to replacing $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ 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).