2020RAA....20..144L

We consider the coplanar planetary four-body problem, where three planets orbit a large star without the cross of their orbits. The system is stable if there is no exchange or cross of orbits. Starting from the Sundman inequality, the equation of the kinematical boundaries is derived. We discuss a reasonable situation, where two planets with known orbits are more massive than the third one. The boundaries of possible motions are controlled by the parameter c²E. If the actual value of c²E is less than or equal to a critical value (c²E)_cr, then the regions of possible motions are bounded and therefore the system is stable. The criteria obtained in special cases are applied to the Solar System and the currently known extrasolar planetary systems. Our results are checked using N-body integrator.