Let us transform to a noninertial frame of reference rotating with angular
velocity about an axis normal
to the orbital plane of masses and , and passing through their center of mass.
It
follows that masses and appear stationary in this new reference frame.
Let us define a Cartesian coordinate system in the rotating frame of reference that is
such that masses and always lie on the axis, and the axis
is parallel to the previously defined axis. It follows that masses
and have the fixed position vectors
and
in our new coordinate system. Finally, let the position vector of
mass be
. See Figure 9.5.
Figure 9.5:
Corotating frame.

According to Section 6.2, the equation of motion of mass in the rotating
reference frame takes the form
where
, and
Here, the second term on the lefthand side of Equation (9.21) is the Coriolis acceleration,
whereas the final term on the righthand side is the centrifugal acceleration. The components of Equation (9.21)
reduce to
which yield
where

(9.30) 
is the sum of the gravitational and centrifugal potentials.
It follows from Equations (9.27)–(9.29) that
Summing the preceding three equations, we obtain

(9.34) 
In other words,

(9.35) 
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 Section 9.9, Exercise 4].
Note, finally, that
the mass is restricted to regions in which

(9.36) 
because
is a positive definite quantity.