# The Jacobi Constant

When dealing with the [Circular Restricted Three-Body Problem](cr3bp.md) 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 {eq}`cr3bp_eom_simp`, by the velocity vector:

:::{math}
:label: 
\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*}
:::

:::{math}

\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

:::{math}
:label: 

\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)

:::

:::{math}
:label: 

\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)**: 

:::{math}
:label: jacobi_constant

\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](cr3bp.md#non-dimensional-circular-restricted-three-body-problem).

Consider the **Jacobian Integral** {eq}`jacobi_constant` with applied normalized potentials $\tilde{V}$ {eq}`cr3bp_V`, and $\tilde{U}$ {eq}`cr3bp_U`. This results in:


:::{math}
:label: jacobi_constant_norm
\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.