The dynamics of new motion styles in the time-dependent four-body problem: weaving periodic solutions

Abstract

In the present investigation, we consider a model originally introduced to obtain new styles of periodic solutions (weaving styles) in the (2, 2)-particle problem. The key feature of these motion styles is the preservation of the axial symmetries of the periodic solutions and the collision free of the bodies. We first derive the equations for the motion of four particles and display how to reduce a dynamical system consisting of twelve differential equations to only three for one body in the new styles of motion, then we prove the existence of collision-free minimizers for the Lagrangian action functional under well chosen class of symmetric loops, to confirm the existence of the weaving periodic solutions for the equal mass four-body problem. Finally, we explore via numerical scheme the dynamics and shapes of the obtained periodic solutions.