August  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
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show all references

References:
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Topol. Methods Nonlinear Anal., 3 (1994), 197-207.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2.  Google Scholar

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Ergodic Theory Dynam. Systems, 23 (2003), 1691-1715. doi: 10.1017/S0143385703000245.  Google Scholar

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in "Proceedings of the International Congress of Mathematicians" III (Beijing, 2002) (Beijing, 2002), Higher Ed. Press, 279-294.  Google Scholar

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New Advances in Celestial Mechanics and Hamiltonian Systems, 63-76, Kluwer/Plenum, New York, (2004).  Google Scholar

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Ann. of Math. (2), 152 (2000), 881-901. doi: 10.2307/2661357.  Google Scholar

[7]

J. Dynam. Differential Equations, 11 (1999), 735-780. doi: 10.1023/A:1022667613764.  Google Scholar

[8]

Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017.  Google Scholar

[9]

in "Ergodic Theory and Dynamical Systems, I (College Park, Md., 1979-80)" 10 of Progr. Math. Birkhäuser Boston, Mass., (1981), 211-333.  Google Scholar

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Arch. Rational Mech. Anal., 179 (2006), 389-412. doi: 10.1007/s00205-005-0396-z.  Google Scholar

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[13]

Invent. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.  Google Scholar

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Invent. Math., 27 (1974), 191-227.  Google Scholar

[15]

Amer. J. Math., 103 (1981), 1323-1341. doi: 10.2307/2374233.  Google Scholar

[16]

Indiana Univ. Math. J., 32 (1983), 221-240. doi: 10.1512/iumj.1983.32.32020.  Google Scholar

[17]

J. Differential Equations, 215 (2005), 1-18. doi: 10.1016/j.jde.2004.11.004.  Google Scholar

[18]

Celestial Mech., 28 (1982), 49-62. doi: 10.1007/BF01230659.  Google Scholar

[19]

Celestial Mech. Dynam. Astronom., 71 (1998/99), 15-33. doi: 10.1023/A:1008397202674.  Google Scholar

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