Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof

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May 2012, 32(5): 1763-1774. doi: 10.3934/dcds.2012.32.1763

Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof

Hsin-Yuan Huang ^1,

1.	School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received December 2010 Revised May 2011 Published January 2012

Abstract
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The Schubart-like orbits in the collinear four-body problem are similar to those discovered numerically by Schubart[12] in the collinear three-body problem. Schubart-like orbits are periodic solutions with exactly two binary collisions and one simultaneous binary collision per period. The proof of the existence of these orbits given in this paper is based on the direct method of Calculus of Variations. We exploit the variational structure of the problem and show that the minimizers of the Lagrangian action functional in a suitably chosen space have the desired properties.

Keywords: variational methods., periodic solutions, Four-body problem.

Mathematics Subject Classification: Primary: 70F16, 70F10; Secondary: 35A1.

Citation: Hsin-Yuan Huang. Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1763-1774. doi: 10.3934/dcds.2012.32.1763

References:

[1]	L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem, Celestial Mechanics and Dynamical Astronomy, 108 (2010), 147-164. doi: 10.1007/s10569-010-9298-y.
[2]	K.-C. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 158 (2001), 293-318. doi: 10.1007/s002050100146.
[3]	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), 881-901. doi: 10.2307/2661357.
[4]	R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99. doi: 10.1016/0022-0396(71)90098-2.
[5]	M. ElBialy, Simultaneous binary collisions in the collinear $N$-body problem, J. Differential Equations, 102 (1993), 209-235. doi: 10.1006/jdeq.1993.1028.
[6]	R. Martínez and C. Simó, The degree of differentiability of the regularization of simultaneous binary collisions in some $N$-body problems, Nonlinearity, 13 (2000), 2107-2130. doi: 10.1088/0951-7715/13/6/312.
[7]	R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175.
[8]	R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 609-620. doi: 10.3934/dcdsb.2008.10.609.
[9]	F. R. Moulton, The straight line solutions of the problem of $n$ bodies, Ann. of Math. (2), 12 (1910), 1-17. doi: 10.2307/2007159.
[10]	T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem, Celestial Mech. Dynam. Astronom., 109 (2011), 229-239. Available from: http://arxiv.org/abs/0811.3199.
[11]	G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems, 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284.
[12]	J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. doi: 10.1002/asna.19562830105.
[13]	M. Sekiguchi and K. Tanikawa, On the symmetric collinear four-body problem, Publication of the Astronomical Society of Japan, 56 (2004), 235-251.
[14]	M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841. doi: 10.1007/s00205-010-0334-6.
[15]	C. Simó and E. Lacomba, Regularization of simultaneous binary collisions in the $n$-body problem, J. Differential Equations, 98 (1992), 241-259. doi: 10.1016/0022-0396(92)90092-2.
[16]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34, Springer-Verlag, Berlin, 2008.
[17]	W. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation, The Restless Universe (Blair Atholl, 2000), Celestial Mech. Dynam. Astronom., 82 (2002), 179-201. doi: 10.1023/A:1014599918133.
[18]	W. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem, Celestial Mech. Dynam. Astronom., 94 (2006), 37-65. doi: 10.1007/s10569-005-2289-8.
[19]	A. Venturelli, "Application de la Minimisation de l'Action au Problḿe des $n$ Corps dans le Plan et dans l'Espace," Thése de Doctorat, Université de Paris 7-Denis Diderot, 2002.
[20]	A. Venturelli, A variational proof of the existence of von Schubart's orbit, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699-717. doi: 10.3934/dcdsb.2008.10.699.
[21]	A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton Mathematical Series, Vol. 5, Princeton University Press, Princeton, N. J., 1941.

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References:

[1]	L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem, Celestial Mechanics and Dynamical Astronomy, 108 (2010), 147-164. doi: 10.1007/s10569-010-9298-y.
[2]	K.-C. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 158 (2001), 293-318. doi: 10.1007/s002050100146.
[3]	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), 881-901. doi: 10.2307/2661357.
[4]	R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99. doi: 10.1016/0022-0396(71)90098-2.
[5]	M. ElBialy, Simultaneous binary collisions in the collinear $N$-body problem, J. Differential Equations, 102 (1993), 209-235. doi: 10.1006/jdeq.1993.1028.
[6]	R. Martínez and C. Simó, The degree of differentiability of the regularization of simultaneous binary collisions in some $N$-body problems, Nonlinearity, 13 (2000), 2107-2130. doi: 10.1088/0951-7715/13/6/312.
[7]	R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175.
[8]	R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 609-620. doi: 10.3934/dcdsb.2008.10.609.
[9]	F. R. Moulton, The straight line solutions of the problem of $n$ bodies, Ann. of Math. (2), 12 (1910), 1-17. doi: 10.2307/2007159.
[10]	T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem, Celestial Mech. Dynam. Astronom., 109 (2011), 229-239. Available from: http://arxiv.org/abs/0811.3199.
[11]	G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems, 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284.
[12]	J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. doi: 10.1002/asna.19562830105.
[13]	M. Sekiguchi and K. Tanikawa, On the symmetric collinear four-body problem, Publication of the Astronomical Society of Japan, 56 (2004), 235-251.
[14]	M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841. doi: 10.1007/s00205-010-0334-6.
[15]	C. Simó and E. Lacomba, Regularization of simultaneous binary collisions in the $n$-body problem, J. Differential Equations, 98 (1992), 241-259. doi: 10.1016/0022-0396(92)90092-2.
[16]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34, Springer-Verlag, Berlin, 2008.
[17]	W. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation, The Restless Universe (Blair Atholl, 2000), Celestial Mech. Dynam. Astronom., 82 (2002), 179-201. doi: 10.1023/A:1014599918133.
[18]	W. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem, Celestial Mech. Dynam. Astronom., 94 (2006), 37-65. doi: 10.1007/s10569-005-2289-8.
[19]	A. Venturelli, "Application de la Minimisation de l'Action au Problḿe des $n$ Corps dans le Plan et dans l'Espace," Thése de Doctorat, Université de Paris 7-Denis Diderot, 2002.
[20]	A. Venturelli, A variational proof of the existence of von Schubart's orbit, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699-717. doi: 10.3934/dcdsb.2008.10.699.
[21]	A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton Mathematical Series, Vol. 5, Princeton University Press, Princeton, N. J., 1941.

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