In this paper, we improve estimations in a previous work  (Arch. Rat. Mech. Anal. 2003) about estimations for the action functional of the $N$-body problem. Our method is based on repeated applications of binary decompositions for the $N$-body system, and is applicable to more general particle systems.
"Estimations for the Action Functional of the $N$-body Problem by Recursive Binary Decompositions." Taiwanese J. Math. 23 (2) 503 - 514, April, 2019. https://doi.org/10.11650/tjm/181008