Taiwanese Journal of Mathematics

Extensions to Chen's Minimizing Equal Mass Parallelogram Solutions

Abdalla Mansur, Daniel Offin, and Alessandro Arsie

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Abstract

In this paper, we study the extension of the minimizing equal mass parallelogram solutions which was derived by Chen in 2001 [2]. Chen's solution was minimizing for one quarter of the period $[0,T]$, where numerical integration had been used in his proof. In this paper we extend Chen's solution in the reduced space to $[0,4T]$ and we show that this extension is also minimizing over the intervals $[0,2T]$ and $[0,4T]$. The minimizing property of the extension is proved without using numerical integration.

Article information

Source
Taiwanese J. Math., Volume 21, Number 6 (2017), 1437-1453.

Dates
Received: 1 July 2017
Revised: 9 October 2017
Accepted: 22 October 2017
First available in Project Euclid: 26 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1508983228

Digital Object Identifier
doi:10.11650/tjm/171003

Mathematical Reviews number (MathSciNet)
MR3732913

Zentralblatt MATH identifier
06871376

Subjects
Primary: 34C35 34C27: Almost and pseudo-almost periodic solutions 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

Keywords
Hamiltonian $n$-body problem equivariant action integral

Citation

Mansur, Abdalla; Offin, Daniel; Arsie, Alessandro. Extensions to Chen's Minimizing Equal Mass Parallelogram Solutions. Taiwanese J. Math. 21 (2017), no. 6, 1437--1453. doi:10.11650/tjm/171003. https://projecteuclid.org/euclid.twjm/1508983228


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References

  • R. Abraham and J. E. Marsden, Foundations of Mechanics, Second edition, Advanced Book Program, Reading, Mass., 1978.
  • K.-C. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal. 158 (2001), no. 4, 293–318.
  • A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math. (2) 152 (2000), no. 3, 881–901.
  • W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99 (1977), no. 5, 961–971.
  • M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, New York, 1966
  • A. Mansur, Instability of Periodic Solutions of Some Rhombus and the Parallelogram Four Body Problem, Ph.D. thesis, Queen's University, 2012.
  • A. Mansur and D. Offin, A minimizing property of homographic solutions, Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 2, 353–360.
  • A. Mansur, D. Offin and M. Lewis, Instability for a family of homographic periodic solutions in the parallelogram four body problem, Qual. Theory Dyn. Syst. 16 (2017), no. 3, 671–688.
  • K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, Second edition, Applied Mathematical Sciences 90, Springer, New York, 2009.
  • R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30
  • S. Zhang and Q. Zhou, A minimizing property of Lagrangian solutions, Acta Math. Sin. (Engl. Ser.) 17 (2001), no. 3, 497–500.
  • ––––, A minimizing property of Eulerian solutions, Celestial Mech. Dynam. Astronom. 90 (2004), no. 3-4, 239–243.