Jacobi integral

In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body 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. x ¨ − 2 n y ˙ = δ U δ x {displaystyle {ddot {x}}-2n{dot {y}}={frac {delta U}{delta x}}}     (1) y ¨ + 2 n x ˙ = δ U δ y {displaystyle {ddot {y}}+2n{dot {x}}={frac {delta U}{delta y}}}     (2) z ¨ = δ U δ z {displaystyle {ddot {z}}={frac {delta U}{delta z}}}     (3) In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body 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. It was named after German mathematician Carl Gustav Jacob Jacobi. One of the suitable coordinate systems used is the so-called synodic or co-rotating system, placed at the barycentre, with the line connecting the two masses μ1, μ2 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 (−μ2, 0) and (+μ1, 0). In the (x, y)-coordinate system, the Jacobi constant is expressed as follows: