Jacobi integral

Consider the function

$\displaystyle C = 2\left(\frac{\mu_1}{\rho_1}+\frac{\mu_2}{\rho_2}\right) + 2\,...
-\skew{3}\dot{\xi}^{\,2}-\skew{3}\dot{\eta}^{\,2}-\skew{3}\dot{\zeta}^{\,2}.$ (9.10)

The time derivative of this function is written

$\displaystyle \dot{C} = - \frac{2\,\mu_1\,\skew{3}\dot{\rho}_1}{\rho_1^{\,2}} -...
- 2\,\skew{3}\dot{\zeta}\,\skew{3}\ddot{\zeta}.$ (9.11)

Moreover, it follows, from Equations (9.3)–(9.4) and (9.8)–(9.9), that

$\displaystyle \rho_1\,\skew{3}\dot{\rho}_1$ $\displaystyle = -(\xi_1\,\skew{3}\dot{\xi}+\eta_1\,\skew{3}\dot{\eta}) + \omega...
+ \eta\,\skew{3}\dot{\eta} + \zeta\,\skew{3}\dot{\zeta},$ (9.12)
$\displaystyle \rho_2\,\skew{3}\dot{\rho}_2$ $\displaystyle = -(\xi_2\,\skew{3}\dot{\xi}+\eta_2\,\skew{3}\dot{\eta}) + \omega...
+ \eta\,\skew{3}\dot{\eta} + \zeta\,\skew{3}\dot{\zeta}.$ (9.13)

Combining Equations (9.5)–(9.7) with the preceding three expressions, after considerable algebra (see Section 9.9, Exercise 1), we obtain

$\displaystyle \frac{dC}{dt} = 0.$ (9.14)

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

We can rearrange Equation (9.10) to give

$\displaystyle {\cal E} \equiv \frac{1}{2}\,(\skew{3}\dot{\xi}^{\,2}+\skew{3}\do...
...{3}\dot{\zeta}^{\,2}) -\left(\frac{\mu_1}{\rho_1}+\frac{\mu_2}{\rho_2}\right)
=$   $\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle \cdot{\bf h} - \frac{C}{2},$ (9.15)

where ${\cal E}$ is the energy (per unit mass) of mass $m_3$, ${\bf h} = {\bf r}\times \dot{\bf r}$ the angular momentum (per unit mass) of mass $m_3$, and $\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.