14.4: Ecuaciones Planetarias de Lagrange

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Pasamos ahora a la Ecuación 14.2.8 para obtener las Ecuaciones Planetarias de Lagrange, que nos permitirán calcular las tasas de cambio de los elementos orbitales si conocemos la forma de la función perturbadora:

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