The dynamics of new motion styles in the time-dependent four-body problem: weaving periodic solutions
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The dynamics of new motion styles in the time-dependent four-body problem: weaving periodic solutions Feras Yousefa
, Osama Alkamb , Ines Sakerc
Department of Mathematics, School of Science, The University of Jordan, Amman 11942, Jordan Received: 16 July 2020 / Accepted: 11 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
1 Introduction In physics, the N -body problem is one of the most important dilemmas for studying of the dynamics of N particles interacting classically with each other. For instance, in celestial mechanics and space mechanics it is used to comprehend how the sun, moon, and other planets move in the solar system. Mathematically, it is a dynamical system of ordinary differential equations which describes the motion of N particles moving under Newton’s laws of motion and Newton’s inverse square law of gravitation. Due to the complexity of the N -body problem, it has been extensively studied in recent years, see for example [1–8]. Strict solutions are known for N = 1 and N = 2 due to the equations that are completely integrable. In 1913, Sundman [9] solved the problem analytically for N = 3 under the assumption that no two particles ever collide and represented the solution of the three-body problem in the form of convergent series. It is worthy of mention that the Sundman’s solution was generalized for N > 3 by Qiu-Dong [10]. In practical terms, the Sundman and Qiu-Dong’s power series expansions of the solutions are not appropriate for solving the N -body problem due to the extremely slow convergence of the series [11]. To the best of our knowledge, the general N -body problem remains completely unsolved for N ≥ 3.
a e-mail: [email protected] (corresponding author) b e-mail: [email protected] c e-mail: [email protected]
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Eur. Phys. J. Plus
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Recently, investigators focus their attention to solve the restricted N -body problem for small values of N , and different types of solutions appeared including the periodic solution. In 1983, Davies, Truman, and Williams (DTW) [12] established a new family of periodic motions for the equal-ma
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