Consider how a particular Lagrange bracket transforms under a rotation of the coordinate system , , about the -axis (if we look along the axis). We can write

(G.41) |

where

(G.42) |

Let the new coordinate system be . A rotation about the -axis though an angle brings the ascending node to the -axis. (See Figure 4.6.) The relation between the old and new coordinates is (see Section A.6)

(G.43) | ||||||

(G.44) | ||||||

and | (G.45) |

The partial derivatives with respect to can be written

(G.46) | ||||||

(G.47) | ||||||

(G.48) | ||||||

and | (G.49) |

where

(G.50) | ||||||

(G.51) | ||||||

(G.52) | ||||||

and | (G.53) |

Let , , , and be the equivalent quantities obtained by replacing by in the preceding equations. It thus follows that

(G.54) | ||||||

and | ||||||

(G.55) |

Hence,

(G.56) |

Now,

(G.57) |

Similarly,

(G.58) |

Let

(G.59) |

Because and , it follows that

(G.60) |

However,

(G.61) |

because the left-hand side is the component of the angular momentum per unit mass parallel to the -axis. Of course, this axis is inclined at an angle to the -axis, which is parallel to the angular momentum vector. Thus, we obtain

Consider a rotation of the coordinate system about the -axis. Let the new coordinate system be , , . A rotation through an angle brings the orbit into the - plane. (See Figure 4.6.) Let

(G.63) |

By analogy with the previous analysis,

(G.64) |

However, and are both zero, because the orbit lies in the - plane. Hence,

Consider, finally, a rotation of the coordinate system about the -axis. Let the final coordinate system be , , . A rotation through an angle brings the perihelion to the -axis. (See Figure 4.6.) Let

(G.66) |

By analogy with the previous analysis,

(G.67) |

However,

(G.68) |

so, from Equations (G.62) and (G.65),

It thus remains to calculate .

The coordinates and --where represents radial distance from the Sun, and is the true anomaly--are functions of the major radius, , the eccentricity, , and the mean anomaly, . Because the Lagrange brackets are independent of time, it is sufficient to evaluate them at ; that is, at the perihelion point. It is easily demonstrated from Equations (4.86) and (4.87) that

(G.70) | ||||||

(G.71) | ||||||

(G.72) | ||||||

and | (G.73) |

at small . Hence, at ,

(G.74) | ||||||

(G.75) | ||||||

(G.76) | ||||||

(G.77) | ||||||

(G.78) | ||||||

and | (G.79) |

because . All other partial derivatives are zero. Because the orbit in the , , coordinate system only depends on the elements , , and , we can write

(G.80) |

Substitution of the values of the derivatives evaluated at into this expression yields

(G.81) | ||

(G.82) | ||

(G.83) |

and

(G.84) |

where . Hence, from Equation (G.69), we obtain