Jacobi integral

It was named after German mathematician Carl Gustav Jacob Jacobi.

Since the first three can be derived from potentials and the last one is perpendicular to the trajectory, they are all conservative, so the energy measured in this system of reference (and hence, the Jacobi integral) is a constant of motion.

In the inertial, sidereal co-ordinate system (ξ, η, ζ), the masses are orbiting the barycentre.

In these co-ordinates the Jacobi constant is expressed by[2] In the co-rotating system, the accelerations can be expressed as derivatives of a single scalar function Using Lagrangian representation of the equations of motion: Multiplying Eqs.

The left side represents the square of the velocity v of the test particle in the co-rotating system.