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Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof
1. | School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States |
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. |
show all references
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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