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November  2016, 36(11): 5971-5991. doi: 10.3934/dcds.2016062

Linear stability of the criss-cross orbit in the equal-mass three-body problem

Xiaojun Chang 1, , Tiancheng Ouyang 2, and  Duokui Yan 3,

1. 

School of Mathematics and Statistics, & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024
2. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602
3. 

School of Mathematics and System Science, Beihang University, Beijing 100191

Received  August 2015 Revised  May 2016 Published  August 2016

In this paper, we study the linear stability of the criss-cross orbit in the planar equal-mass three-body problem. In each period of the criss-cross orbit, the configurations of three masses are switching from a straight line to an isosceles triangle eight times. By analyzing its symmetry properties and variational characterization, we show that the criss-cross orbit is linearly stable via index theory.
Keywords: linear stability, periodic orbit, Maslov-type index., Celestial mechanics.
Mathematics Subject Classification: Primary: 70F07; Secondary: 70F10, 53D1.
Citation: Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062
References:
[1]

V. Barutello, R. Jadanza and A. Portaluri, Linear instability of relative equilibria for n-body problems in the plane, J. Diff. Eqn., 257 (2014), 1773-1813. doi: 10.1016/j.jde.2014.05.017.

[2]

V. Barutello, R. Jadanza and A. Portaluri, Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory, Arch. Ration. Mech. Anal., 219 (2016), 387-444. doi: 10.1007/s00205-015-0898-2.

[3]

R. Broucke, On relative periodic solutions of the planar general three-body problem, Celestial Mech., 12 (1975), 439-462. doi: 10.1007/BF01595390.

[4]

S. Cappell, R. Lee and E. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186. doi: 10.1002/cpa.3160470202.

[5]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901. doi: 10.2307/2661357.

[6]

A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing 2002), 279-294, Higher Ed. Press, Beijing, 2002.

[7]

K.-C. Chen, T. Ouyang and Z. Xia, Action-minimizing periodic and quasi-periodic solutions in the n-body problem, Math. Res. Lett., 19 (2012), 483-497. doi: 10.4310/MRL.2012.v19.n2.a19.

[8]

K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problems with various choices of masses, Ann. of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325.

[9]

K.-C. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Commun. Math. Phys., 291 (2009), 403-441. doi: 10.1007/s00220-009-0769-5.

[10]

D. Ferrario and S. Terracini, On the existence of collisionless equivalent minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.

[11]

M. Hénon, A family of periodic orbits of the planar three-body problem, and their atability, Celest. Mech. Dynam. Astron., 13 (1976), 267-285. doi: 10.1007/BF01228647.

[12]

X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian system with application to Figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y.

[13]

X. Hu and S. Sun, Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem, Adv. Math., 223 (2010), 98-119. doi: 10.1016/j.aim.2009.07.017.

[14]

X. Hu, Y. Long and S. Sun, Linear stability of elliptic Lagrangian solutions of the classical planar three-body problem via index theory, Arch. Ration. Mech. Anal., 213 (2014), 993-1045. doi: 10.1007/s00205-014-0749-6.

[15]

T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Berlin-Heidelberg-New York, Spring-Verlag, 1995.

[16]

Y. Long, Index Theory For Symplectic Paths With Applications, Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.

[17]

C. Marchal, How the method of minimization of action avoids singularities, Celestial Mech. Dynam. Astronom., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.

[18]

C. Moore, Braids in classical gravity, Phys. Rev. Lett., 70 (1993), 3675-3679. doi: 10.1103/PhysRevLett.70.3675.

[19]

C. Moore, , http://www.santafe.edu/textasciitilde moore/gallery.html., (). 

[20]

C. Moore and M. Nauenberg, New periodic orbits for the n-body problem, J. of Comput. Nonlin. Dyn., 1 (2006), 307-311. doi: 10.1115/1.2338323.

[21]

T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76. doi: 10.1016/j.physd.2015.05.015.

[22]

G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dynam. Sys., 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284.

[23]

C. Zhu, A generalized Morse index theorem, In Analysis, Geometry and Topology of Elliptic Operators, Hackensack, NJ: World Sci. Publ. (2006), 493-540.

[24]

G. Zhu and Y. Long, Linear stability of some symplectic matrices, Front. Math. China, 5 (2010), 361-368. doi: 10.1007/s11464-010-0008-6.

show all references

References:
[1]

V. Barutello, R. Jadanza and A. Portaluri, Linear instability of relative equilibria for n-body problems in the plane, J. Diff. Eqn., 257 (2014), 1773-1813. doi: 10.1016/j.jde.2014.05.017.

[2]

V. Barutello, R. Jadanza and A. Portaluri, Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory, Arch. Ration. Mech. Anal., 219 (2016), 387-444. doi: 10.1007/s00205-015-0898-2.

[3]

R. Broucke, On relative periodic solutions of the planar general three-body problem, Celestial Mech., 12 (1975), 439-462. doi: 10.1007/BF01595390.

[4]

S. Cappell, R. Lee and E. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186. doi: 10.1002/cpa.3160470202.

[5]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901. doi: 10.2307/2661357.

[6]

A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing 2002), 279-294, Higher Ed. Press, Beijing, 2002.

[7]

K.-C. Chen, T. Ouyang and Z. Xia, Action-minimizing periodic and quasi-periodic solutions in the n-body problem, Math. Res. Lett., 19 (2012), 483-497. doi: 10.4310/MRL.2012.v19.n2.a19.

[8]

K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problems with various choices of masses, Ann. of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325.

[9]

K.-C. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Commun. Math. Phys., 291 (2009), 403-441. doi: 10.1007/s00220-009-0769-5.

[10]

D. Ferrario and S. Terracini, On the existence of collisionless equivalent minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.

[11]

M. Hénon, A family of periodic orbits of the planar three-body problem, and their atability, Celest. Mech. Dynam. Astron., 13 (1976), 267-285. doi: 10.1007/BF01228647.

[12]

X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian system with application to Figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y.

[13]

X. Hu and S. Sun, Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem, Adv. Math., 223 (2010), 98-119. doi: 10.1016/j.aim.2009.07.017.

[14]

X. Hu, Y. Long and S. Sun, Linear stability of elliptic Lagrangian solutions of the classical planar three-body problem via index theory, Arch. Ration. Mech. Anal., 213 (2014), 993-1045. doi: 10.1007/s00205-014-0749-6.

[15]

T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Berlin-Heidelberg-New York, Spring-Verlag, 1995.

[16]

Y. Long, Index Theory For Symplectic Paths With Applications, Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.

[17]

C. Marchal, How the method of minimization of action avoids singularities, Celestial Mech. Dynam. Astronom., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.

[18]

C. Moore, Braids in classical gravity, Phys. Rev. Lett., 70 (1993), 3675-3679. doi: 10.1103/PhysRevLett.70.3675.

[19]

C. Moore, , http://www.santafe.edu/textasciitilde moore/gallery.html., (). 

[20]

C. Moore and M. Nauenberg, New periodic orbits for the n-body problem, J. of Comput. Nonlin. Dyn., 1 (2006), 307-311. doi: 10.1115/1.2338323.

[21]

T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76. doi: 10.1016/j.physd.2015.05.015.

[22]

G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dynam. Sys., 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284.

[23]

C. Zhu, A generalized Morse index theorem, In Analysis, Geometry and Topology of Elliptic Operators, Hackensack, NJ: World Sci. Publ. (2006), 493-540.

[24]

G. Zhu and Y. Long, Linear stability of some symplectic matrices, Front. Math. China, 5 (2010), 361-368. doi: 10.1007/s11464-010-0008-6.

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