Action minimizing orbits in the trapezoidal four body problem

In this paper, we study the minimizing property for the isosceles trapezoid solutions of the four-body problem. We prove that the minimizers of the action functional restricted to homographic solutions are the Keplerian elliptical solutions, and this functional has a minimum equal to $\frac{3}{2}(2\pi {)}^{2/3}{T}^{1/3}{\left(\frac{\xi (a,b)}{\eta (a,b)}\right)}^{2/3}$. Further, we investigate the dynamical behavior in the trapezoidal four-body problem using the Poincaré surface of section method.