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Jacobi integral
Consider the function

(9.10) 
The time derivative of this function is
written

(9.11) 
Moreover, it follows, from Equations (9.3)(9.4) and (9.8)(9.9),
that
Combining Equations (9.5)(9.7) with the preceding three expressions, after considerable
algebra (see Exercise 1), we obtain

(9.14) 
In other words, the function
which is usually referred to as the Jacobi integralis a
constant of the motion.
We can rearrange Equation (9.10) to give
where
is the energy (per unit mass) of mass
,
the angular momentum
(per unit mass) of mass
, and
the orbital angular velocity of the other two masses.
Note, however, that
is not a constant of the motion. Hence,
is not
a constant of the motion either. In fact, the Jacobi integral is the only constant of the
motion in the circular restricted threebody problem. Incidentally, the energy
of mass
is not a conserved quantity because the other two masses in the system
are moving.
Next: Tisserand criterion
Up: Threebody problem
Previous: Circular restricted threebody problem
Richard Fitzpatrick
20160331