Dynamics of the the dihedral four-body problem

Davide L. Ferrario; Alessandro Portaluri

doi:10.3934/dcdss.2013.6.925

Article Contents

2013, Volume 6, Issue 4: 925-974. Doi: 10.3934/dcdss.2013.6.925

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Dynamics of the the dihedral four-body problem

Davide L. Ferrario^1,2, and
Alessandro Portaluri^3,4,5,

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: August 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 70F15; Secondary: 70F10, 70F16.

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References

[1]	Topol. Methods Nonlinear Anal., 3 (1994), 197-207.
[2]	Arch. Ration. Mech. Anal., 170 (2003), 247-276.doi: 10.1007/s00205-003-0277-2.
[3]	Ergodic Theory Dynam. Systems, 23 (2003), 1691-1715.doi: 10.1017/S0143385703000245.
[4]	in "Proceedings of the International Congress of Mathematicians" III (Beijing, 2002) (Beijing, 2002), Higher Ed. Press, 279-294.
[5]	New Advances in Celestial Mechanics and Hamiltonian Systems, 63-76, Kluwer/Plenum, New York, (2004).
[6]	Ann. of Math. (2), 152 (2000), 881-901.doi: 10.2307/2661357.
[7]	J. Dynam. Differential Equations, 11 (1999), 735-780.doi: 10.1023/A:1022667613764.
[8]	Invent. Math., 60 (1980), 249-267.doi: 10.1007/BF01390017.
[9]	in "Ergodic Theory and Dynamical Systems, I (College Park, Md., 1979-80)" 10 of Progr. Math. Birkhäuser Boston, Mass., (1981), 211-333.
[10]	Arch. Rational Mech. Anal., 179 (2006), 389-412.doi: 10.1007/s00205-005-0396-z.
[11]	Adv. in Math., 2 (2007), 763-784.doi: 10.1016/j.aim.2007.01.009.
[12]	Nonlinearity, 21 (2008), 1-15.doi: 10.1088/0951-7715/21/6/009.
[13]	Invent. Math., 155 (2004), 305-362.doi: 10.1007/s00222-003-0322-7.
[14]	Invent. Math., 27 (1974), 191-227.
[15]	Amer. J. Math., 103 (1981), 1323-1341.doi: 10.2307/2374233.
[16]	Indiana Univ. Math. J., 32 (1983), 221-240.doi: 10.1512/iumj.1983.32.32020.
[17]	J. Differential Equations, 215 (2005), 1-18.doi: 10.1016/j.jde.2004.11.004.
[18]	Celestial Mech., 28 (1982), 49-62.doi: 10.1007/BF01230659.
[19]	Celestial Mech. Dynam. Astronom., 71 (1998/99), 15-33.doi: 10.1023/A:1008397202674.