EFFECTIVE POTENTIAL, HILL-TYPE STABILITY AND SUNDMAN INEQUALITY

The velocity decomposition of Saari (Celest. Mech. 33, 299, 1984) is s hown to capture the best possible effective potential of the general N -body problem. This improves the effective potential inherent to Sundm an's inequality. Saari's approach is simplified by virtue of the inert ia tensor and inertia ellipsoid. The inequality stronger than Sundman' s obtained for the flat problem by Saari (Celest. Mech. 40, 197, 1987) is not only extended to the general N-body problem but is also shown to be valid under more relaxed conditions. For the spatial three-body problem, we also obtain a more explicit expression of the inequality w hich was used to plot the Hill-type surfaces by Ge and Leng (Celest. M ech. 53,233, 1992). It is proved that Hill-type stability guarantees o ne of the hierarchical stability conditions, and prevents cross-over o f orbits.