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Jacobi Integral
Consider the function

(1039) 
The time derivative of this function is
written

(1040) 
Moreover, it follows, from Equations (1032)(1033) and (1037)(1038),
that
Combining Equations (1034)(1036) with the above three expressions, we obtain (after considerable
algebra)

(1043) 
In other words, the function which is usually referred to as the Jacobi integralis a constant of the motion.
Now, we can rearrange Equation (1039) to give

(1044) 
where is the energy (per unit mass) of mass ,
its angular momentum
(per unit 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: The ThreeBody Problem
Previous: Circular Restricted ThreeBody Problem
Richard Fitzpatrick
20110331