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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   (1041)   (1042)

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 integral--is 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 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: The Three-Body Problem Previous: Circular Restricted Three-Body Problem
Richard Fitzpatrick 2011-03-31