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# 14.4: Lagrange's Planetary Equations

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We now go to Equation 14.2.8 to obtain Lagrange’s Planetary Equations, which will enable us to calculate the rates of change of the orbital elements if we know the form of the perturbing function:

\begin{align} \dot{a} &= - \frac{2a^2}{GMm} \frac{\partial R}{\partial T} , \label{14.4.1} \\[5pt] \dot{e} &= - \frac{a(1-e^2)}{GMme} \frac{\partial R}{\partial T} , \label{14.4.2} \\[5pt] i &= - \frac{1}{\sqrt{GMm^2 a (1-e^2) \sin i}} \frac{\partial R}{\partial \Omega} - \frac{1}{me} \sqrt{\frac{1 - e^2}{GMa}} \frac{\partial R}{\partial ω} , \label{14.4.3} \\[5pt] \dot{ω} &= \frac{1}{me} \sqrt{\frac{1 - e^2}{GMa}} \frac{\partial R}{\partial e} - \frac{1}{\sqrt{GMm^2 a (1 - e^2)} \tan i } \frac{\partial R}{\partial i} , \label{14.4.4} \\[5pt] \dot{Ω} &= \frac{1}{\sqrt{GMm^2 (1 - e^2) \sin i}} \frac{\partial R}{\partial i} , \label{14.4.5} \\[5pt] \dot{T} &= \frac{2a^2}{GMm} \frac{\partial R}{\partial a} + \frac{a(1 - e^2)}{GMme} \frac{\partial R}{\partial e}. \label{14.4.6} \end{align}

14.4: Lagrange's Planetary Equations 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 conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.