Jacobi Integral

Consider the function

$$C = 2\left(\frac{\mu_1}{\rho_1}+\frac{\mu_2}{\rho_2}\right) ... ...eta}-\eta\,\dot{\xi}) -\dot{\xi}^2-\dot{\eta}^2-\dot{\zeta}^2.$$ (1039)

The time derivative of this function is written

$$\dot{C} = - \frac{2\,\mu_1\,\dot{\rho}_1}{\rho_1^{\,2}} - \f... ...} - 2\,\dot{\eta}\,\ddot{\eta} - 2\,\dot{\zeta}\,\ddot{\zeta}.$$ (1040)

Moreover, it follows, from Equations (1032)-(1033) and (1037)-(1038), that

$$\rho_1\,\dot{\rho}_1 = -(\xi_1\,\dot{\xi}+\eta_1\,\dot{\eta}) + \omega\,(\xi\,\eta_1-\eta\,\xi_1) + \xi\,\dot{\xi} + \eta\,\dot{\eta} + \zeta\,\dot{\zeta},$$ (1041)

$$\rho_2\,\dot{\rho}_2 = -(\xi_2\,\dot{\xi}+\eta_2\,\dot{\eta}) + \omega\,(\xi\,\eta_2-\eta\,\xi_2) + \xi\,\dot{\xi} + \eta\,\dot{\eta} + \zeta\,\dot{\zeta}.$$ (1042)

Combining Equations (1034)-(1036) with the above three expressions, we obtain (after considerable algebra)

$$\dot{C} = 0.$$ (1043)

In other words, the function $C$--which is usually referred to as the Jacobi integral--is a constant of the motion.

Now, we can rearrange Equation (1039) to give

$${\cal E} \equiv \frac{1}{2}\,(\dot{\xi}^2+\dot{\eta}^2+\dot{... ...}\right) = \mbox{\boldmath$\omega$}\cdot{\bf h} - \frac{C}{2},$$ (1044)

where ${\cal E}$ is the energy (per unit mass) of mass $m_3$, ${\bf h} = {\bf r}\times \dot{\bf r}$ its angular momentum (per unit mass), and $\mbox{\boldmath$\omega$}=(0,\,0,\,\omega)$ the orbital angular velocity of the other two masses. Note, however, that ${\bf h}$ is not a constant of the motion. Hence, ${\cal E}$ 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 $m_3$ is not a conserved quantity because the other two masses in the system are moving.