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Next: Tisserand Criterion Up: The Three-Body Problem Previous: Circular Restricted Three-Body Problem


Jacobi Integral

Consider the function
\begin{displaymath}
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.
\end{displaymath} (1039)

The time derivative of this function is written
\begin{displaymath}
\dot{C} = - \frac{2\,\mu_1\,\dot{\rho}_1}{\rho_1^{\,2}} - \f...
...} - 2\,\dot{\eta}\,\ddot{\eta}
- 2\,\dot{\zeta}\,\ddot{\zeta}.
\end{displaymath} (1040)

Moreover, it follows, from Equations (1032)-(1033) and (1037)-(1038), that
$\displaystyle \rho_1\,\dot{\rho}_1$ $\textstyle =$ $\displaystyle -(\xi_1\,\dot{\xi}+\eta_1\,\dot{\eta}) + \omega\,(\xi\,\eta_1-\eta\,\xi_1) + \xi\,\dot{\xi}
+ \eta\,\dot{\eta} + \zeta\,\dot{\zeta},$ (1041)
$\displaystyle \rho_2\,\dot{\rho}_2$ $\textstyle =$ $\displaystyle -(\xi_2\,\dot{\xi}+\eta_2\,\dot{\eta}) + \omega\,(\xi\,\eta_2-\eta\,\xi_2) + \xi\,\dot{\xi}
+ \eta\,\dot{\eta} + \zeta\,\dot{\zeta}.$ (1042)

Combining Equations (1034)-(1036) with the above three expressions, we obtain (after considerable algebra)
\begin{displaymath}
\dot{C} = 0.
\end{displaymath} (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

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


next up previous
Next: Tisserand Criterion Up: The Three-Body Problem Previous: Circular Restricted Three-Body Problem
Richard Fitzpatrick 2011-03-31