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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.

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Barrabés, E., Cors, J. M., Pinyol, C. and Soler, J., ‘Hip-hop solutions of the $2n$ -body problem’, Celestial Mech. Dynam. Astronom. 95 (2006), 5566.
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), 189226.
Barutello, V. and Terracini, S., ‘Action minimizing orbits in the $n$ -body problem with simple choreography constraint’, Nonlinearity 17 (2004), 20152039.
Bessis, D., Digne, F. and Michel, J., ‘Springer theory in braid groups and the Birman–Ko–Lee monoid’, Pacific J. Math. 205(2) (2002), 287309.
Chenciner, A., ‘A note by Poincaré’, Regul. Chaotic Dyn. 10(2) (2005), 119128.
Chenciner, A., ‘Poincaré and the three-body problem’, Sémin. Poincaré XVI (2012), 45133.
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), 287308.
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), 881901.
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), 139152.
Fadell, E. and Neuwirth, L., ‘Configuration spaces’, Math. Scand. 10 (1962), 111118.
Farb, B. and Margalit, D., A Primer on Mapping Class Groups, Princeton Mathematical Series, 49 (Princeton University Press, Princeton, NJ, 2012).
Ferrario, D., ‘Transitive decomposition of symmetry groups for the $n$ -body problem’, Adv. Math. 213(2) (2007), 763784.
Ferrario, D. L. and Terracini, S., ‘On the existence of collisionless equivariant minimizers for the classical $n$ -body problem’, Invent. Math. 155(2) (2004), 305362.
Fox, R. and Neuwirth, L., ‘The braid groups’, Math. Scand. 10 (1962), 119126.
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), 283332.
Golubitsky, M. and Stewart, I., ‘Hopf bifurcation in the presence of symmetry’, Arch. Ration. Mech. Anal. 87 (1985), 107165.
González-Meneses, J., ‘The $n\mathrm{th} $ root of a braid is unique up to conjugacy’, Algebr. Geom. Topol. 3 (2003), 11031118, electronic.
González-Meneses, J. and Ventura, E., (2011) Twisted conjugacy in braid groups. ArXiv 1104.5690 [MATH.GT].
González-Meneses, J. and Wiest, B., ‘On the structure of the centralizer of a braid’, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), 729757.
Gordon, W. B., ‘Conservative dynamical systems involving strong forces’, Trans. Amer. Math. Soc. 204 (1975), 113135.
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)].
Hatcher, A., Algebraic Topology (Cambridge University Press, Cambridge, 2002).
Jost, J. and Li-Jost, X., Calculus of Variations, Cambridge Studies in Advanced Mathematics, 64 (Cambridge University Press, Cambridge, 1998).
Kassel, C. and Turaev, V., Braid Groups, Graduate Texts in Mathematics, vol. 247 (Springer, New York, 2008).
Klingenberg, W., Lectures on Closed Geodesics, Grundlehren der Mathematischen Wissenschaften, 230 (Springer, Berlin, 1978).
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), 482493.
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), 237293.
Montgomery, R., ‘The $N$ -body problem, the braid group, and action-minimizing periodic solutions’, Nonlinearity 11 (1998), 363376.
Montgomery, R., ‘A new solution to the three-body problem’, Not. Amer. Math. Soc. 48 (2001), 471481.
Moore, C., ‘Braids in classical dynamics’, Phys. Rev. Lett. 70(24) (1993), 36753679.
Palais, R., ‘The principle of symmetric criticality’, Comm. Math. Phys. 69(1) (1979), 1930.
Rhodes, F., ‘On the fundamental group of a transformation group’, Proc. Lond. Math. Soc. (3) 16 (1966), 635650.
Simó, C., ‘New families of solutions in $n$ -body problems’, in European Congress of Mathematics, Vol. I (2001).
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), 265284.
Spanier, E. H., Algebraic Topology (Springer, New York, 1981), Corrected reprint.
Steckles, K., Loop Spaces and Choreographies in Dynamical Systems. Ph.D. Thesis, University of Manchester, 2011.
Stewart, I., ‘Symmetry methods in collisionless many-body problems’, J. Nonlinear Sci. 6 (1996), 543563.
Terracini, S., ‘On the variational approach to the periodic $n$ -body problem’, Celestial Mech. Dynam. Astronom. 95 (2006), 325.
Vershinin, V. V., ‘Braid groups and loop spaces’, Uspekhi Mat. Nauk 54(2(326)) (1999), 384.
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