Jacobi integral

Remarks

$C$ is a first integral: the equations of motion are $x'' - 2 y' = \Omega_x$, $y'' + 2 x' = \Omega_y$, $z'' = \Omega_z$, and multiplying them by $x'$, $y'$, $z'$ and adding gives $x' x'' + y' y'' + z' z'' = \frac{d}{dt} \Omega$, so $C' = 0$. Example: a third body at rest at the triangular Lagrange point $\left( \frac{1}{2} - \mu, \frac{\sqrt{3}}{2}, 0 \right)$ has $r_1 = r_2 = 1$, and its Jacobi constant is $C = \frac{1}{4} - \mu + \mu^2 + \frac{3}{4} + 2 = 3 - \mu + \mu^2$. The ordinary energy $\frac{1}{2}(x'^2 + y'^2 + z'^2) - \frac{1 - \mu}{r_1} - \frac{\mu}{r_2}$ in the rotating frame is not a first integral, because it omits the centrifugal term $\frac{x^2 + y^2}{2}$.

Since $x'^2 + y'^2 + z'^2 \ge 0$, a motion with Jacobi constant $C$ stays in the Hill region $\set{2 \Omega \ge C}$. Its boundary $\set{2 \Omega = C}$ is the zero-velocity surface. In units where the distance, the angular velocity and the gravitational parameters are not normalised, $C = n^2 (x^2 + y^2) + 2 \left( \frac{G m_1}{r_1} + \frac{G m_2}{r_2} \right) - (x'^2 + y'^2 + z'^2)$, where $n$ is the angular velocity of the primaries.