According to Section 6.2, the equation of motion of mass in the rotating reference frame takes the form

where

(9.22) | ||

(9.23) |

Here, the second term on the left-hand side of Equation (9.21) is the Coriolis acceleration, whereas the final term on the right-hand side is the centrifugal acceleration. The components of Equation (9.21) reduce to

(9.24) | ||||||

(9.25) | ||||||

and | (9.26) |

which yield

where

is the sum of the gravitational and centrifugal potentials.

It follows from Equations (9.27)-(9.29) that

(9.31) | ||||||

(9.32) | ||||||

and | (9.33) |

Summing the preceding three equations, we obtain

(9.34) |

In other words,

is a constant of the motion, where . In fact, is the Jacobi integral introduced in Section 9.3 [it is easily demonstrated that Equations (9.10) and (9.35) are identical; see Exercise 4]. Note, finally, that the mass is restricted to regions in which

(9.36) |

because is a positive definite quantity.