Kozai Lidov Cycles = Simple Pendulum
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: $e_{\text{max}}^2-e^2=h=l\left(1-\cosθ\right)$. 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 $\boldsymbolα=\textbf{j}+\textbf{e}, \boldsymbolβ=\textbf{j}-\textbf{e}$ 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 $\boldsymbolα$ and $\boldsymbolβ$ simplify to $\dot{\boldsymbolα}=2\partial_{\boldsymbolα} ϕ\times \boldsymbolα$ and $\dot{\boldsymbolβ}=2\partial_{\boldsymbolβ} ϕ\times \boldsymbolβ$, 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 $\boldsymbolα^2=\boldsymbolβ^2=1$, and they are distributed uniformly and independently on the unit sphere for a uniform distribution in phase space (with a fixed energy).