(G.30) |

The new equations can be written in a more compact form via the introduction of

(G.31) |

where , , and . Thus, the new equations become

for . Note, incidentally, that

(G.33) | ||||||

and | (G.34) |

Let

(G.35) |

where and are any two orbital elements. It follows that

(G.36) |

or

(G.37) |

However, in the preceding expression, and stand for coordinates and velocities of Keplerian orbits calculated with treated as constants. Thus, we can write and , giving

because

(G.39) |

where . Expression (G.38) reduces to

(G.40) |

where . Hence, we conclude that Lagrange brackets are functions of the osculating orbital elements, , but are not explicit functions of . It follows that we can evaluate these brackets at any convenient point in the orbit.