Abstract: The two-center problem, also known as Euler’s three-body problem, is a classic example of integrable systems. Among its periodic solutions, planetary type solutions are periodic solutions which enclose both centers. Inspired by advances on n-body and n-center problems via variational techniques developed during the past two decades, a recent paper (Arch. Rat. Mech. Ana. 2022) shows the minimizing property of planetary type solutions for any given masses of centers at fixed positions, as long as the period is above a mass-dependent threshold value. In this paper, we provide further discussions regarding this minimizing approach. In particular, we improve the above-mentioned mass-dependent threshold value by refining estimates for action values.

Key words: wo-center problem, n-body problem, variational method, collision singularity