The Jacobi Constant

When dealing with the Circular Restricted Three-Body Problem there exists a time-invariant integral of motion known as the Jacobian Integral of Motion. To get this integral of motion, multiple the Circular Restricted Three-Body Problem equations of motion, Equation (16), by the velocity vector:

(1)\[\begin{split}\begin{align*} \dot{x}\left(\ddot{x} - 2n\dot{y}\right) &= \dot{x}\left(\frac{\partial{V}}{\partial{x}}\right) \\ \dot{y}\left(\ddot{y} + 2n\dot{x}\right) &= \dot{y}\left(\frac{\partial{V}}{\partial{y}}\right) \\ \dot{z}\left(\ddot{z} \right) &= \dot{z}\left(\frac{\partial{V}}{\partial{z}}\right) \end{align*}\end{split}\]

\[\dot{x}\ddot{x} + \dot{y}\ddot{y} + \dot{z}\ddot{z} = \frac{\partial{V}}{\partial{x}}\dot{x} + \frac{\partial{V}}{\partial{y}}\dot{y} + \frac{\partial{V}}{\partial{z}}\dot{z} \]

and then integrate both ends

(2)\[\frac{1}{2} \frac{d}{dt} \left[ \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right] = \frac{d}{dt} V\left(x,y,z \right)\]

(3)\[\frac{d}{dt} \left[ \frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right) - V \right] = 0\]

This yields the constant Jacobi Integral (J):

(4)\[\frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right) - V\left(x,y,z \right) = J \]

Non-Dimensional Jacobi Constant

At the same time there exists a Non-Dimensional Jacobi Constant that follows the same formation as the Non-Dimensional Circular Restricted Three-Body Problem.

Consider the Jacobian Integral (4) with applied normalized potentials \(\tilde{V}\) (14), and \(\tilde{U}\) (2). This results in:

(5)\[\frac{1}{2}\left( {\dot{x}^*}^2 + {\dot{y}^*}^2 + {\dot{z}^*}^2 \right) - \tilde{V}\left(x^*,y^*,z^*\right) = \tilde{J} \]

which is still a constant.