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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 which 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 48.
Figure 48:
The corotating frame.

According to Chapter 7, the equation of motion of mass in the rotating
reference frame takes the form

(1050) 
where
, and
Here, the second term on the lefthand side of Equation (1050) is the Coriolis acceleration,
whereas the final term on the righthand side is the centrifugal acceleration. The components of Equation (1050)
reduce to
which yield
where

(1059) 
is the sum of the gravitational and centrifugal potentials.
Now, it follows from Equations (1056)(1058) that
Summing the above three equations, we obtain

(1063) 
In other words,

(1064) 
is a constant of the motion, where
. In fact, is the
Jacobi integral introduced in Section 13.3 [it is easily demonstrated that Equations (1039) and
(1064) are identical].
Note, finally, that
the mass is restricted to regions in which

(1065) 
since is a positive definite quantity.
Next: Lagrange Points
Up: The ThreeBody Problem
Previous: Tisserand Criterion
Richard Fitzpatrick
20110331