## Taiwanese Journal of Mathematics

### Estimations for the Action Functional of the $N$-body Problem by Recursive Binary Decompositions

Kuo-Chang Chen

#### Abstract

In this paper, we improve estimations in a previous work [3] (Arch. Rat. Mech. Anal. 2003) about estimations for the action functional of the $N$-body problem. Our method is based on repeated applications of binary decompositions for the $N$-body system, and is applicable to more general particle systems.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 2 (2019), 503-514.

Dates
Revised: 1 October 2018
Accepted: 8 October 2018
First available in Project Euclid: 22 October 2018

https://projecteuclid.org/euclid.twjm/1540195380

Digital Object Identifier
doi:10.11650/tjm/181008

Mathematical Reviews number (MathSciNet)
MR3936010

Subjects
Primary: 70F10: $n$-body problems

#### Citation

Chen, Kuo-Chang. Estimations for the Action Functional of the $N$-body Problem by Recursive Binary Decompositions. Taiwanese J. Math. 23 (2019), no. 2, 503--514. doi:10.11650/tjm/181008. https://projecteuclid.org/euclid.twjm/1540195380

#### References

• V. Barutello and S. Terracini, Action minimizing orbits in the $n$-body problem with simple choreography constraint, Nonlinearity, 17 (2004), no. 6, 2015–2039.
• K.-C. Chen, On Chenciner-Montgomery's orbit in the three-body problem, Discrete Contin. Dynam. Systems 7 (2001), no. 1, 85–90.
• ––––, Binary decompositions for planar $N$-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal. 170 (2003), no. 3, 247–276.
• ––––, Removing collision singularities from action minimizers for the $N$-body problem with free boundaries, Arch. Ration. Mech. Anal. 181 (2006), no. 2, 311–331.
• ––––, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Ann. of Math. (2) 167 (2008), no. 2, 325–348.
• K.-C. Chen and Y.-C. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys. 291 (2009), no. 2, 403–441.
• K.-C. Chen, T. Ouyang and Z. Xia, Action-minimizing periodic and quasi-periodic solutions in the $n$-body problem, Math. Res. Lett. 19 (2012), no. 2, 483–497.
• A. Chenciner, Action minimizing solutions in the Newtonian $n$-body problem: from homology to symmetry, Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol III, 279–294.
• ––––, Action minimizing periodic orbits in the Newtonian $n$-body problem, in: Celestial Mechanics (Evanston, IL, 1999), 71–90, Contemp. Math. 292, Amer. Math. Soc., Providence, RI, 2002.
• ––––, Some facts and more questions about the Eight, in: Topological Methods, Variational Methods and Their Applications (Taiyuan, 2002), 77–88, World Sci. Publ., River Edge, NJ, 2003.
• 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.
• A. Chenciner, J. Féjoz and R. Montgomery, Rotating eights I: The three $\Gamma_{i}$ families, Nonlinearity 18 (2005), no. 3, 1407–1424.
• A. Chenciner and A. Venturelli, Minima de l'intégrale d'action du problème newtonien de 4 corps de masses égales dans $\mathbb{R}^3$: orbites “hip-hop”, Celestial Mech. Dynam. Astronomy 77 (2000), no. 2, 139–152.
• D. L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math. 155 (2004), no. 2, 305–362.
• H. Fukuda, T. Fujiwara and H. Ozaki, Figure-eight choreographies of the equal mass three-body problem with Lennard-Jones-type potentials, J. Phys. A: Math. Theor. 50 (2017), no. 10, 105202, 16 pp.
• G. Fusco, G. F. Gronchi and P. Negrini, Platonic polyhedra, topological constraints and periodic solutions of the classical $N$-body problem, Invent. Math. 185 (2011), no. 2, 283–332.
• W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99 (1977), no. 5, 961–971.
• R. Moeckel, Shooting for the eight: a topological existence proof for a figure-eight orbit of the three-body problem, in: Differential Equations: Geometry, Symmetries and Integrability, 287–310, Abel Symp. 5, Springer, Berlin, 2009.
• R. Montgomery, $N$-body choreographies, Scholarpedia 5 (2010), no. 11, 10666.
• C. Moore, Braids in classical dynamics, Phys. Rev. Lett. 70 (1993), no. 24, 3675– 3679.
• R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30.
• M. Shibayama, Variational proof of the existence of the super-eight orbit in the four-body problem, Arch. Ration. Mech. Anal. 214 (2014), no. 1, 77–98.
• N. Soave and S. Terracini, Symbolic dynamics for the $N$-centre problem at negative energies, Discrete Contin. Dyn. Syst. 32 (2012), no. 9, 3245–3301.
• S. Terracini, $n$-body and choreographies, in: Mathematics of Complexity and Dynamical Systems Vols. 1-3, 1043–1069, Springer, New York, 2012.
• S. Terracini and A. Venturelli, Symmetric trajectories for the $2N$-body problem with equal masses, Arch. Ration. Mech. Anal. 184 (2007), no. 3, 465–493.
• G. Yu, Simple choreographies of the planar Newtonian $N$-body problem, Arch. Ration. Mech. Anal. 225 (2017), no. 2, 901–935.
• S. Zhang and Q. Zhou, Variational methods for the choreography solution to the three-body problem, Sci. China Ser. A 45 (2002), no. 5, 594–597.