With the Jacobi coordinates r, R, reduced masses m, , and the ratio z≥r/R, the equations of motion and the energy of the system are written by
The functions A(z) and B(z) satisfy a relation A(z)z3≷B(z) according as z≷k*, where z=k* corresponds to the central configurations. In the region z≤k* the sub-energies vary monotonically for every binary collision point(r=0) in succession (rectilinear case and isosceles case), and for every point in succession with maximal value of r (rectilinear case), provided that Ŕ>0 is satisfied throughout.
The motion is escape under the following initial conditions:
I. Rectilinear Case:
i) Ŕ>0 and h2>0, at a point with z≤k* and a maximal value of r;
ii) Ŕ>0, h2>0, and h1<0, at a point with z=k*;
iii) Ŕ>0, ŕ<0, h2>0, and h1<0, at a point with z≥k*;
II. Isosceles Case: Ŕ>0, h2>0, and h1<0, at a binary collision point (r=0).
Finally, in the rectilinear case if Ŕ>0 and h2<0 at a binary collision point(r=0), then the trajectory in the (R,r)–space gets out of the region z≤k*, otherwise the motion is ejection without escape.