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

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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.

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References

  1. J. Cronin, Some periodic solutions of a four-body problem. ICARUS 3, 423–428 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  2. M.G. Crandall, Two families of periodic solutions of the plane four-body problem. Am. J. Math. 16, 275–318 (1967)

    Article  MathSciNet  Google Scholar 

  3. A. Albouy, The symmetric central configurations of four equal masses. Contemp. Math. 198, 131–136 (1996)

    Article  MathSciNet  Google Scholar 

  4. K.C. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses. Arch. Rat. Mech. Anal. 158(4), 293–318 (2001)

    Article  MathSciNet  Google Scholar 

  5. G. Spurgin, Periodic orbits in the four-body problem with large and small masses. J. Differ. Equ. 249, 1131–1144 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  6. A.N. Baltagiannis, K.E. Papadakis, Families of periodic orbits in the restricted four-body problem. Ast. Sp. Sci. 336(2), 357–367 (2011)

    Article  ADS  Google Scholar 

  7. A. Alsaedi, F. Yousef, S. Bushnaq, S. Momani, New styles of periodic solutions of the classical six-body problem. Math. Comput. Simul. 159, 183–196 (2019)

    Article  MathSciNet  Google Scholar 

  8. E.I. Abouelmagd, J.L.G. Guirao, A.K. Pal, Periodic solution of the nonlinear Sitnikov restricted three-body problem. N. Astron. 75, 101319 (2020). https://doi.org/10.1016/j.newast.2019.101319

    Article  Google Scholar 

  9. K.F. Sundman, Memoire sur le probleme des trois corp. Acta Math. 36(1), 105–179 (1913)

    Article  MathSciNet  Google Scholar 

  10. W. Qiu-Dong, The global solution of the \(n\)-body problem. Celes. Mech. Dyn. Astron. 50(1), 73–88 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  11. D. Belorizky, Sur la solution du probleme des trois corps, donnee par M. Sundman. CR Acad. Sci. 193, 766–768 (1931)

    MATH  Google Scholar 

  12. I.M. Davies, A. Truman, D. Williams, Classical periodic solutions of the equal mass \(2n\)-body problem, \(2n\)-ion problem, and the \(n\)-electron atom problem. Phys. Lett. 99A, 15–18 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  13. A. Chenciner, R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. 152(3), 881–901 (2000)

    Article  MathSciNet  Google Scholar 

  14. A.E. Al-Saedi, The Orbits of Periodic Solutions of Many Body Problems (University of Wales Swansea, Thesis, 2001)

    Google Scholar 

  15. G. Hardy, J. Littlewood, G. Polya, Inequalities, 2nd edn. (Cambridge University Press, Cambridge, 1952)

    MATH  Google Scholar 

  16. H. Brezis, Functional Analysis (Sobolev Spaces and Partial Differential Equations. Universitext) (Springer, Berlin, 2011)

    MATH  Google Scholar 

  17. R. Adams, J. Fournierm, Sobolev Spaces, 2nd edn. (Academic Press, London, 2003)

    Google Scholar 

  18. M.G. Cui, Y.Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Hilbert Space (Nova Science, New York, 2009)

    MATH  Google Scholar 

  19. M.G. Sakar, Iterative reproducing kernel Hilbert spaces method for Riccati differential equations. J. Comput. Appl. Math. 309, 163–174 (2017)

    Article  MathSciNet  Google Scholar 

  20. J. Niu, M.Q. Xu, Y.Z. Lin, Q. Xue, Numerical solution of nonlinear singular boundary value problems. J. Comput. Appl. Math. 331, 42–51 (2018)

    Article  MathSciNet  Google Scholar 

  21. B. Maayah, F. Yousef, O. Abu Arqub, S. Momani, A. Alsaedi, Computing bifurcations behavior of mixed type singular time-fractional partial integrodifferential equations of Dirichlet functions types in Hilbert space with error analysis. Filomat 33(12), 3845–3853 (2019)

    Article  MathSciNet  Google Scholar 

  22. H. Li, W. Jiang, W.Y. Li, Space-time spectral method for the Cattaneo equation with time fractional derivative. Appl. Math. Comput. 349, 325–336 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for the careful reading of this manuscript and for the thoughtful comments and constructive suggestions, which helped to improve the quality of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, The University of Jordan, Amman, 11942, Jordan

    Feras Yousef, Osama Alkam & Ines Saker

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  1. Feras Yousef
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  2. Osama Alkam
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  3. Ines Saker
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Correspondence to Feras Yousef.

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Yousef, F., Alkam, O. & Saker, I. The dynamics of new motion styles in the time-dependent four-body problem: weaving periodic solutions. Eur. Phys. J. Plus 135, 742 (2020). https://doi.org/10.1140/epjp/s13360-020-00774-1

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