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:
[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

show all references

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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