Action minimizing orbits in the trapezoidal four body problem

Abstract

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.

Keywords

• central configuration
• four-body problem
• minimizers
• trapezoidal problem
• action functional
• homographic solutions