Alternative forms of Lagrange planetary equations

It can be seen, from Equations (G.115)–(G.120), that in the limit of small eccentricity, $e$, and small inclination, $I$, certain terms on the right-hand sides of the Lagrange planetary equations become singular. This problem can be alleviated by defining the alternative orbital elements,

$$h =e\,\sin\varpi,$$ (G.121)

$$k = e\,\cos\varpi,$$ (G.122)

$$p = \sin I\,\sin{\mit\Omega},$$ (G.123)

$$q = \sin I\,\cos{\mit\Omega}.$$ (G.124)

If we write the Lagrange planetary equations in terms of these new elements then we obtain

$$\frac{da}{dt} =\frac{2}{n\,a}\,\frac{\partial {\cal R}}{\partial\skew{5}\bar{\lambda}},\displaybreak[0]$$ (G.125)

$$\frac{d\skew{5}\bar{\lambda}}{dt} =n - \frac{2}{n\,a}\,\frac{\partial {\cal R}}{\partial a}+ \frac{... ...\partial {\cal R}}{\partial h} + k\,\frac{\partial {\cal R}}{\partial k}\right)$$

$$\phantom{=}+ \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2}... ...}{\partial p} + q\,\frac{\partial {\cal R}}{\partial q}\right),\displaybreak[0]$$ (G.126)

$$\frac{dh}{dt} =-\frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,[1+(1-e^{\,2})^{1/2}]}\, h... ...}} + \frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}}\,\frac{\partial {\cal R}}{\partial k}$$

$$\phantom{=}+ \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2}... ...partial {\cal R}}{\partial p} + q\,\frac{\partial {\cal R}}{\partial q}\right),$$ (G.127)

$$\frac{dk}{dt} = -\frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,[1+(1-e^{\,2})^{1/2}]}\,k... ...a}} -\frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}}\,\frac{\partial {\cal R}}{\partial h}$$

$$\phantom{=}- \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2}... ...partial {\cal R}}{\partial p} + q\,\frac{\partial {\cal R}}{\partial q}\right),$$ (G.128)

$$\frac{dp}{dt} = - \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2})^{1/2}}\... ...rac{\partial{\cal R}}{\partial h}-h\,\frac{\partial{\cal R}}{\partial k}\right)$$

$$\phantom{=}+ \frac{\cos I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal R}}{\partial q},$$ (G.129)

$$\frac{dq}{dt} =- \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2})^{1/2}}\,... ...rac{\partial{\cal R}}{\partial h}-h\,\frac{\partial{\cal R}}{\partial k}\right)$$

$$\phantom{=}- \frac{\cos I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal R}}{\partial p}.$$ (G.130)

Note that the new equations now contain no singular terms in the limit $e, I\rightarrow 0$.

It is sometimes convenient to write the Lagrange planetary equations in terms of the mean anomaly, ${\cal M} = \skew{5}\bar{\lambda}-\varpi$, and the argument of the perigee, $\omega=\varpi-{\mit\Omega}$, rather than $\skew{5}\bar{\lambda}$ and $\omega$. Making the appropriate substitutions, the equations take the form (Brouwer and Clemence 1961)

$$\frac{da}{dt} = \frac{2}{n\,a}\,\frac{\partial{\cal R}}{\partial{\cal M}},$$ (G.131)

$$\frac{d{\cal M}}{dt} = n -\frac{2}{n\,a}\,\frac{\partial {\cal R}}{\partial a}- \frac{1-e^{\,2}}{n\,a^{\,2}\,e}\,\frac{\partial{\cal R}}{\partial e},$$ (G.132)

$$\frac{de}{dt} =\frac{1-e^{\,2}}{n\,a^{\,2}\,e}\,\frac{\partial {\cal R}}{\parti... ...ac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,e}\,\frac{\partial{\cal R}}{\partial \omega},$$ (G.133)

$$\frac{dI}{dt} = \frac{\cot I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial{\c... ...2})^{-1/2}}{n\,a^{\,2}\,\sin I}\,\frac{\partial{\cal R}}{\partial{\mit\Omega}},$$ (G.134)

$$\frac{d\omega}{dt} = \frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,e}\,\frac{\partial {\cal R... ...ac{\cot I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial{\cal R}}{\partial I},$$ (G.135)

$$\frac{d{\mit\Omega}}{dt} = \frac{(1-e^{\,2})^{-1/2}}{n\,a^{\,2}\,\sin I}\,\frac{\partial{\cal R}}{\partial I}.$$ (G.136)