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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  (9.12) and  (9.13)

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 integral--is a constant of the motion.

We can rearrange Equation (9.10) to give   (9.15)

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 three-body 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: Three-body problem Previous: Circular restricted three-body problem
Richard Fitzpatrick 2016-03-31