On the Principle of Least Interaction Action and the Laplacean satellites of Jupiter and Uranus

The Principle of Least Interaction Action, which explains the observed preference in the Solar System for two-satellite resonant configurations, is shown to apply also to the Laplacean satellites of Jupiter and Uranus, in the sense that these triplet resonant structures lie close to configurations for which the time-mean of the action associated with the mutual interaction of the satellites is an overall minimum. Far from the minimum configuration, significant changes take place in the major semi-axes on time-scales ～10⁶-10⁷y. Both systems require times ～10⁸ y to come close to the minimum configuration; to approach resonance to the observed precision of the Laplace relationship (3×10^?4 for the Uranian case, 2×10^?7 for the Jovian case) requires, for both systems, a time closely comparable with the age of the solar system.