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.


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:

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


  • <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&nbsp;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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