# American Institute of Mathematical Sciences

April  2013, 6(4): 925-974. doi: 10.3934/dcdss.2013.6.925

## Dynamics of the the dihedral four-body problem

 1 Deptartment of Mathematics and Applications, University of Milano-Bicocca 2 Via R. Cozzi, 53, 20125 Milano 3 Dipartimento di Matematica e Fisica Ennio De Giorgi 4 Universit del Salento 5 73100 Lecce

Received  October 2011 Revised  April 2012 Published  December 2012

Consider four point particles with equal masses in the euclideanspace,subject to the following symmetry constraint: at each instant theyare symmetric with respect to the dihedral group $D_2$,that is the groupgenerated by two rotations of angle $\pi$ around twoorthogonal axes.Under ahomogeneous potential of degree $-\alpha$ for $0<\alpha<2$,this is a subproblem of the four-body problem,inwhich all orbits have zero angular momentum and the configurationspace is three-dimensional.In this paper westudy the flow in McGehee coordinates on the collision manifold,anddiscuss the qualitative behavior of orbits which reach or come close to a total collision.
Citation: Davide L. Ferrario, Alessandro Portaluri. Dynamics of the the dihedral four-body problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 925-974. doi: 10.3934/dcdss.2013.6.925
##### References:
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##### References:
 [1] A. Ambrosetti and V. Coti Zelati, Non-collision periodic solutions for a class of symmetric $3$-body type problems,, Topol. Methods Nonlinear Anal., 3 (1994), 197.   Google Scholar [2] K.-C. Chen, Binary decompositions for planar $N$-body problems and symmetric periodic solutions,, Arch. Ration. Mech. Anal., 170 (2003), 247.  doi: 10.1007/s00205-003-0277-2.  Google Scholar [3] K.-C. Chen, Variational methods on periodic and quasi-periodic solutions for the $N$-body problem,, Ergodic Theory Dynam. Systems, 23 (2003), 1691.  doi: 10.1017/S0143385703000245.  Google Scholar [4] A. Chenciner, Action minimizing solutions of the Newtonian $n$-body problem: from homology to symmetry,, in, III (2002), 279.   Google Scholar [5] A. Chenciner, Are there perverse choreographies?,, New Advances in Celestial Mechanics and Hamiltonian Systems, (2004), 63.   Google Scholar [6] 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), 881.  doi: 10.2307/2661357.  Google Scholar [7] J. Delgado and C. Vidal, The tetrahedral {$4$-body problem},, J. Dynam. Differential Equations, 11 (1999), 735.  doi: 10.1023/A:1022667613764.  Google Scholar [8] R. L. Devaney, Triple collision in the planar isosceles three-body problem,, Invent. Math., 60 (1980), 249.  doi: 10.1007/BF01390017.  Google Scholar [9] R. L. Devaney, Singularities in classical mechanical systems,, in, 10 (1981), 1979.   Google Scholar [10] D. L. Ferrario, Symmetry groups and non-planar collisionless action-minimizing solutions of the three-body problem in three-dimensional space,, Arch. Rational Mech. Anal., 179 (2006), 389.  doi: 10.1007/s00205-005-0396-z.  Google Scholar [11] D. L. Ferrario, Transitive decomposition of symmetry groups for the $n$-body problem,, Adv. in Math., 2 (2007), 763.  doi: 10.1016/j.aim.2007.01.009.  Google Scholar [12] D. L. Ferrario and A. Portaluri, On th dihedral $n$-body problem,, Nonlinearity, 21 (2008), 1.  doi: 10.1088/0951-7715/21/6/009.  Google Scholar [13] D. L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem,, Invent. Math., 155 (2004), 305.  doi: 10.1007/s00222-003-0322-7.  Google Scholar [14] R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.   Google Scholar [15] R. Moeckel, Orbits of the three-body problem which pass infinitely close to triple collision,, Amer. J. Math., 103 (1981), 1323.  doi: 10.2307/2374233.  Google Scholar [16] R. Moeckel, Orbits near triple collision in the three-body problem,, Indiana Univ. Math. J., 32 (1983), 221.  doi: 10.1512/iumj.1983.32.32020.  Google Scholar [17] M. Salomone and Z. Xia, Non-planar minimizers and rotational symmetry in the $N$-body problem,, J. Differential Equations, 215 (2005), 1.  doi: 10.1016/j.jde.2004.11.004.  Google Scholar [18] C. Simó and E. Lacomba, Analysis of some degenerate quadruple collisions,, Celestial Mech., 28 (1982), 49.  doi: 10.1007/BF01230659.  Google Scholar [19] C. Vidal, The tetrahedral $4$-body problem with rotation,, Celestial Mech. Dynam. Astronom., 71 (), 15.  doi: 10.1023/A:1008397202674.  Google Scholar
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