Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T02:43:19.828Z Has data issue: false hasContentIssue false
  • English
  • Français

CLASSIFICATION OF SYMMETRY GROUPS FOR PLANAR $n$-BODY CHOREOGRAPHIES

Part of: Variational problems in infinite-dimensional spaces Smooth dynamical systems: general theory

Published online by Cambridge University Press:  13 December 2013

JAMES MONTALDI  and
KATRINA STECKLES
JAMES MONTALDI
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, [email protected]
KATRINA STECKLES
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the $n$-body problem: periodic motions where the $n$ bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part, we classify all possible symmetry groups of planar $n$-body collision-free choreographies. These symmetry groups fall into two infinite families and, if $n$ is odd, three exceptional groups. In the second part, we develop the equivariant fundamental group and use it to determine the topology of the space of loops with a given symmetry, which we show is related to certain cosets of the pure braid group in the full braid group, and to centralizers of elements of the corresponding coset. In particular, we refine the symmetry classification by classifying the connected components of the set of loops with any given symmetry. This leads to the existence of many new choreographies in $n$-body systems governed by a strong force potential.

MSC classification

Secondary: 37C80: Symmetries, equivariant dynamical systems 70F10: $n$-body problems 58E40: Group actions
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 1 , 2013 , e5
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2013.

References

Barrabés, E., Cors, J. M., Pinyol, C. and Soler, J., ‘Hip-hop solutions of the $2n$ -body problem’, Celestial Mech. Dynam. Astronom. 95 (2006), 55–66.CrossRefGoogle Scholar
Barutello, V., Ferrario, D. and Terracini, S., ‘Symmetry groups of the planar 3-body problem and action-mimizing trajectories’, Arch. Ration. Mech. Anal. 190 (2008), 189–226.CrossRefGoogle Scholar
Barutello, V. and Terracini, S., ‘Action minimizing orbits in the $n$ -body problem with simple choreography constraint’, Nonlinearity 17 (2004), 2015–2039.CrossRefGoogle Scholar
Bessis, D., Digne, F. and Michel, J., ‘Springer theory in braid groups and the Birman–Ko–Lee monoid’, Pacific J. Math. 205(2) (2002), 287–309.CrossRefGoogle Scholar
Chenciner, A., ‘A note by Poincaré’, Regul. Chaotic Dyn. 10(2) (2005), 119–128.CrossRefGoogle Scholar
Chenciner, A., ‘Poincaré and the three-body problem’, Sémin. Poincaré XVI (2012), 45–133.Google Scholar
Chenciner, A., Gerver, J., Montgomery, R. and Simó, C., ‘Simple choreographic motions of $N$ bodies: a preliminary study’, in Geometry, Mechanics, and Dynamics (Springer, New York, 2002), 287–308.CrossRefGoogle Scholar
Chenciner, A. and Montgomery, R., ‘A remarkable periodic solution of the three-body problem in the case of equal masses’, Ann. of Math. (2) 152 (2000), 881–901.CrossRefGoogle Scholar
Chenciner, A. and Venturelli, A., ‘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. Astronom. 77(2) (2000), 139–152.CrossRefGoogle Scholar
Fadell, E. and Neuwirth, L., ‘Configuration spaces’, Math. Scand. 10 (1962), 111–118.CrossRefGoogle Scholar
Farb, B. and Margalit, D., A Primer on Mapping Class Groups, Princeton Mathematical Series, 49 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Ferrario, D., ‘Transitive decomposition of symmetry groups for the $n$ -body problem’, Adv. Math. 213(2) (2007), 763–784.CrossRefGoogle Scholar
Ferrario, D. L. and Terracini, S., ‘On the existence of collisionless equivariant minimizers for the classical $n$ -body problem’, Invent. Math. 155(2) (2004), 305–362.CrossRefGoogle Scholar
Fox, R. and Neuwirth, L., ‘The braid groups’, Math. Scand. 10 (1962), 119–126.CrossRefGoogle Scholar
Fusco, G., Gronchi, G. F. and Negrini, P., ‘Platonic polyhedra, topological constraints and periodic solutions of the classical $N$ -body problem’, Invent. Math. 185(2) (2011), 283–332.CrossRefGoogle Scholar
Golubitsky, M. and Stewart, I., ‘Hopf bifurcation in the presence of symmetry’, Arch. Ration. Mech. Anal. 87 (1985), 107–165.CrossRefGoogle Scholar
González-Meneses, J., ‘The $n\mathrm{th} $ root of a braid is unique up to conjugacy’, Algebr. Geom. Topol. 3 (2003), 1103–1118, electronic.CrossRefGoogle Scholar
González-Meneses, J. and Ventura, E., (2011) Twisted conjugacy in braid groups. ArXiv 1104.5690 [MATH.GT].Google Scholar
González-Meneses, J. and Wiest, B., ‘On the structure of the centralizer of a braid’, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), 729–757.CrossRefGoogle Scholar
Gordon, W. B., ‘Conservative dynamical systems involving strong forces’, Trans. Amer. Math. Soc. 204 (1975), 113–135.CrossRefGoogle Scholar
Guillemin, V. W. and Sternberg, S., ‘Supersymmetry and equivariant de Rham theory’, in Mathematics Past and Present (Springer, Berlin, 1999), With an appendix containing two reprints by Henri Cartan [MR0042426 (13,107e); MR0042427 (13,107f)].Google Scholar
Hatcher, A., Algebraic Topology (Cambridge University Press, Cambridge, 2002).Google Scholar
Jost, J. and Li-Jost, X., Calculus of Variations, Cambridge Studies in Advanced Mathematics, 64 (Cambridge University Press, Cambridge, 1998).Google Scholar
Kassel, C. and Turaev, V., Braid Groups, Graduate Texts in Mathematics, vol. 247 (Springer, New York, 2008).CrossRefGoogle Scholar
Klingenberg, W., Lectures on Closed Geodesics, Grundlehren der Mathematischen Wissenschaften, 230 (Springer, Berlin, 1978).CrossRefGoogle Scholar
McCord, C., Montaldi, J., Roberts, M. and Sbano, L., ‘Relative periodic orbits of symmetric Lagrangian systems’, in EQUADIFF 2003 (World Sci. Publ, Hackensack, NJ, 2005), 482–493.CrossRefGoogle Scholar
Montaldi, J., http://www.maths.manchester.ac.uk/jm/Choreographies.Google Scholar
Montaldi, J. A., Roberts, R. M. and Stewart, I. N., ‘Periodic solutions near equilibria of symmetric Hamiltonian systems’, Philos. Trans. R. Soc. Lond. Ser. A 325 (1988), 237–293.Google Scholar
Montgomery, R., ‘The $N$ -body problem, the braid group, and action-minimizing periodic solutions’, Nonlinearity 11 (1998), 363–376.CrossRefGoogle Scholar
Montgomery, R., ‘A new solution to the three-body problem’, Not. Amer. Math. Soc. 48 (2001), 471–481.Google Scholar
Moore, C., ‘Braids in classical dynamics’, Phys. Rev. Lett. 70(24) (1993), 3675–3679.CrossRefGoogle ScholarPubMed
Palais, R., ‘The principle of symmetric criticality’, Comm. Math. Phys. 69(1) (1979), 19–30.CrossRefGoogle Scholar
Rhodes, F., ‘On the fundamental group of a transformation group’, Proc. Lond. Math. Soc. (3) 16 (1966), 635–650.CrossRefGoogle Scholar
Simó, C., ‘New families of solutions in $n$ -body problems’, in European Congress of Mathematics, Vol. I (2001).Google Scholar
Simó, C., ‘Periodic orbits of the planar $n$ -body problem with equal masses and all bodies on the same path’, in The Restless Universe (eds. Steves, B. A. and Maciejewski, A. J.) (2001), 265–284.Google Scholar
Spanier, E. H., Algebraic Topology (Springer, New York, 1981), Corrected reprint.CrossRefGoogle Scholar
Steckles, K., Loop Spaces and Choreographies in Dynamical Systems. Ph.D. Thesis, University of Manchester, 2011. http://eprints.ma.man.ac.uk/1807.Google Scholar
Stewart, I., ‘Symmetry methods in collisionless many-body problems’, J. Nonlinear Sci. 6 (1996), 543–563.CrossRefGoogle Scholar
Terracini, S., ‘On the variational approach to the periodic $n$ -body problem’, Celestial Mech. Dynam. Astronom. 95 (2006), 3–25.CrossRefGoogle Scholar
Vershinin, V. V., ‘Braid groups and loop spaces’, Uspekhi Mat. Nauk 54(2(326)) (1999), 3–84.Google Scholar
Supplementary material: File

Montaldi and Steckles Supplementary Material

Supplementary Material

Download Montaldi and Steckles Supplementary Material(File)
File 207.1 KB