Lagrange planetary equations

(G.86) | ||||||

(G.87) | ||||||

and | (G.88) |

with . Hence,

(G.89) | ||||||

(G.90) | ||||||

(G.91) | ||||||

(G.92) | ||||||

(G.93) | ||||||

and | (G.94) |

with all other partial derivatives zero. Thus, from Equation (G.85), the only non-zero Lagrange brackets are

(G.95) | ||||||

(G.96) | ||||||

(G.97) | ||||||

(G.98) | ||||||

(G.99) | ||||||

and | (G.100) |

Hence, Equations (G.32) yield

(G.101) | ||||

(G.102) | ||||

(G.103) | ||||

(G.104) | ||||

(G.105) | ||||

and | (G.106) |

Finally, Equations (G.95)-(G.106) can be rearranged to give

Equations (G.107)-(G.112), which specify the time evolution of the osculating orbital elements of our planet under the action of the disturbing function, are known collectively as the

In fact, the orbital element always appears in the disturbing function in the combination . This combination is known as the mean longitude, and is denote . It follows that

(G.113) | ||||||

and | (G.114) |

The integral appearing in the previous equation is problematic. Fortunately, it can easily be eliminated by replacing the variable by . In this case, the Lagrange planetary equations become

where is taken at constant , and at constant (Brouwer and Clemence 1961).