Abstract
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.
Highlights
To understand the dynamics presented by a total collision of the masses or the equilibrium state of a rotating system, we are led to the concept of central configurations
E four-body problem can be considered from two different perspectives. e perturbative approach where we study the dynamical aspects as a perturbation of the threebody dynamics and assume that one of the masses is vanishingly small, or the global approach where we allow the masses to vary independently and stay positive
We prove that the minimizers for the action functional restricted to the homographic solutions are the Keplerian elliptical solutions for the four-body problem with three equal and unequal masses
Summary
To understand the dynamics presented by a total collision of the masses or the equilibrium state of a rotating system, we are led to the concept of central configurations. We prove that the minimizers for the action functional restricted to the homographic solutions are the Keplerian elliptical solutions for the four-body problem with three equal and unequal masses. Celli [34] proves the existence of planar diamond and trapezoidal central configurations for two pairs of equal masses.
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