Kozai Lidov Cycles = Simple Pendulum
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915213646757888 |
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| author | Basha, Roi D. Klein, Ygal Y. Katz, Boaz |
| author_facet | Basha, Roi D. Klein, Ygal Y. Katz, Boaz |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kozai Lidov Cycles = Simple Pendulum Basha, Roi D. Klein, Ygal Y. Katz, Boaz Solar and Stellar Astrophysics Earth and Planetary Astrophysics Classical Physics 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). |
| title | Kozai Lidov Cycles = Simple Pendulum |
| topic | Solar and Stellar Astrophysics Earth and Planetary Astrophysics Classical Physics |
| url | https://arxiv.org/abs/2503.19972 |