Skip to main content
Physics LibreTexts

14.3: The Poisson Brackets for the Orbital Elements

  • Page ID
    6877
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    A worked example is in order. From Equations 14.2.7 and 14.2.8, we see that the Poisson brackets are defined by

    \[\{ A_i , A_k \}_{α_j , β_j} = \sum_j \left( \frac{\partial A_i}{\partial α_j} \frac{\partial A_k}{\partial β_j} - \frac{\partial A_i}{\partial β_j} \frac{\partial A_k}{\partial α_j} \right) . \label{14.3.1} \]

    The \(A_i\) are the orbital elements.

    For our example, we shall calculate \(\{ Ω , i \}\) and we write out the sum in full:

    \[\begin{align} \{ Ω , i \} &= \sum_j \left( \frac{\partial Ω}{\partial α_j} \frac{\partial i}{\partial β_j} - \frac{\partial Ω}{\partial β_j} \frac{\partial i}{\partial α_j} \right) \\[4pt] &= \frac{\partial Ω}{\partial α_1} \frac{\partial i}{\partial β_1} + \frac{\partial Ω}{\partial α_2} \frac{\partial i}{\partial β_2} + \frac{\partial Ω}{\partial α_3} \frac{\partial i}{\partial β_3} - \frac{\partial Ω}{\partial β_1} \frac{\partial i}{\partial α_1} - \frac{\partial Ω}{\partial β_2} \frac{\partial i}{\partial α_2} - \frac{\partial Ω}{\partial β_3} \frac{\partial i}{\partial α_3}. \label{14.3.2} \end{align}\]

    Refer now to Equations 10.11.27 and 29, and we find

    \[\{ Ω, i \} = 0 + 0 + 0 - 0 + \frac{1}{α_3 \sqrt{1 - α_2^2 / α_3^2}} - 0 . \label{14.3.3} \]

    Finally, referring to Equations 10.11.20 and 21, we obtain

    \[ \{ Ω , i \} = \frac{1}{\sqrt{GMm^2 a (1-e^2). \sin i}} . \label{14.3.4}\]

    Proceeding in a similar manner for the others, we obtain

    \[\begin{align} \{ a , T \} &= - \frac{2a^2}{GMm}, \label{14.3.5} \\[4pt] \{ e , T \} &= - \frac{a(1-e^2)}{GMme}, \label{14.3.6} \\[4pt] \{ i , ω \} &= \frac{1}{\sqrt{GMm^2 a (1-e^2). \tan i}} . \label{14.3.7} \\[4pt] \{ e , ω \} &= - \frac{\sqrt{1-e^2}}{em \sqrt{GMa}}, \label{14.3.8} \end{align}\]

    In addition, we have, of course,

    \[ \{ i , Ω \} = - \{ Ω , i \} , \{ T , a \} = - \{ a , T \} , \{ T , e \} = - \{ e , T \} \text{ and } \{ ω , i \} = - \{ i , ω \} .\]

    All other pairs are zero.


    This page titled 14.3: The Poisson Brackets for the Orbital Elements is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.