Jacobi Integral

# Jacobi integral

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Description:
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 (&minus;μ<sub>2</sub>,&nbsp;0) and (+μ<sub>1</sub>,&nbsp;0)<sup>1</sup>.

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

[itex]C_J=n^2 (x^2+y^2) + 2 left(frac+fracright) - left(dot x^2+dot y^2+dot z^2right)[/itex]

where:

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

Note that the Jacobi integral is minus twice...

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