Jacobi Integral

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In celestial mechanics, **Jacobi's integral** (named after Carl Gustav Jacob Jacobi) is the only known conserved quantity for the restricted three-body problem problem ; unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible. The integral has been used to derive numerous solutions in special cases.

## Definition

### Synodic system

One of the suitable coordinates system used is so called *synodic* or co-rotating system, placed at the barycentre, with the line connecting the two masses *μ*<sub>1</sub>, *μ*<sub>2</sub> chosen as *x*-axis and the length unit equal to their distance. As the system co-rotates with the two masses, they remain **stationary** and positioned at (−*μ*<sub>2</sub>, 0) and (+*μ*<sub>1</sub>, 0)<sup>1</sup>.

In the (*x*, *y*)-coordinate system, the Jacobi constant is expressed as follows:

where:

Note that the Jacobi integral is minus twice...

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In the (

- <math>C_J=n^2 (x^2+y^2) + 2 left(frac+fracright) - left(dot x^2+dot y^2+dot z^2right)</math>

where:

- <math>n=frac</math> is the mean motion (orbital period T)
- <math>mu_1=Gm_1,!,mu_2=Gm_2,!</math>, for the two masses
*m*<sub>1</sub>,*m*<sub>2</sub> and the gravitational constant *G* - <math>r_1,!,r_2,!</math> are distances of the test particle from the two masses

Note that the Jacobi integral is minus twice...

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