In the current article, we study the kite four-body problems with the goal of identifying global regions in the mass parameter space which admits a corresponding central configuration of the four masses. We consider two different types of symmetrical configurations. In each of the two cases, the existence of a continuous family of central configurations for positive masses is shown. We address the dynamical aspect of periodic solutions in the settings considered and show that the minimizers of the classical action functional restricted to the homographic solutions are the Keplerian elliptical solutions. Finally, we provide numerical explorations via Poincaré cross-sections, to show the existence of periodic and quasiperiodic solutions within the broader dynamical context of the four-body problem.

中文翻译：

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风筝四体问题的中心配置和动作最小化轨道

在当前的文章中，我们研究风筝四体问题，目的是确定质量参数空间中的全局区域，该全局区域允许四个质量的相应中心配置。我们考虑两种不同类型的对称配置。在这两种情况中的每一种情况下，都显示出存在一个连续的正质量的中心构型族。我们在考虑的环境中解决了周期解的动力学问题，并证明了限于同形解的经典作用函数的极小值是开普勒椭圆解。最后，我们通过Poincaré横截面进行数值探索，以证明四体问题在更宽泛的动力学范围内存在周期和拟周期解。