They correspond to places where the pseudo-potential is locally flat. libration points) are equilibrium points in the rotating frame. This routine assumes that G = 1 and the distance between the primary objects in 1. lagrangePoints.m - returns a matrix containing the (x,y,z) coordinates of the five Lagrange points for given values of m1 and m2.crtbpPotential.m - returns the pseudo-potential for the circular, restricted three-body.Requires crtbpPotential.m and lagrangePoints.m to run. Also marks the positions of the Lagrange points with "+" symbols. crtbpZeroVel.m - Plots zero-velocity curves for difference values of the Jacobi integral.This plot also show the positions of the five Lagrange points (see next section). Therefore a particle with this value of CJ can transition back and forth between orbits around each object. However, the zero-velocity curve for CJ = 3.92 encompasses both m1 and m2.
For example, if a particle with CJ = 4 is initially in orbit around the green planet, it will be stuck there forever (unless it is given a velocity boost by some means). The zero-velocity curves bound the shaded 'forbidden' regions where a particle with the specified Jacobi integral can not venture. The following figure shows zero velocity curves for different Jacobi integrals.
Such curves are called zero-velocity curves are are equivalent to turning points for potential wells in inertial frames of reference.
For a given Jacobi integral, one can calculate the curve in space where the velocity would go to zero. This property may be exploited to place bounds on the particle's motion. The Jacobi integral for a particle will remain constant as it orbits the system. The form of the Jacobi integral is similar to the total energy: it has two terms, one a pseudo-potential and the other a quadratic velocity term like the kinetic energy.