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

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.
Citation: Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.jde.2014.05.017.  Google Scholar

[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.  doi: 10.1007/s00205-015-0898-2.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.2307/2661357.  Google Scholar

[6]

A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry,, Proceedings of the International Congress of Mathematicians, (2002), 279.   Google Scholar

[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.  doi: 10.4310/MRL.2012.v19.n2.a19.  Google Scholar

[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.  doi: 10.4007/annals.2008.167.325.  Google Scholar

[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.  doi: 10.1007/s00220-009-0769-5.  Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1007/s00220-009-0860-y.  Google Scholar

[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.  doi: 10.1016/j.aim.2009.07.017.  Google Scholar

[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.  doi: 10.1007/s00205-014-0749-6.  Google Scholar

[15]

T. Kato, Perturbation theory for linear operators,, Classics in Mathematics, (1995).   Google Scholar

[16]

Y. Long, Index Theory For Symplectic Paths With Applications,, Birkhäuser Verlag, (2002).  doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

C. Zhu, A generalized Morse index theorem,, In Analysis, (2006), 493.   Google Scholar

[24]

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

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.  doi: 10.1016/j.jde.2014.05.017.  Google Scholar

[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.  doi: 10.1007/s00205-015-0898-2.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.2307/2661357.  Google Scholar

[6]

A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry,, Proceedings of the International Congress of Mathematicians, (2002), 279.   Google Scholar

[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.  doi: 10.4310/MRL.2012.v19.n2.a19.  Google Scholar

[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.  doi: 10.4007/annals.2008.167.325.  Google Scholar

[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.  doi: 10.1007/s00220-009-0769-5.  Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1007/s00220-009-0860-y.  Google Scholar

[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.  doi: 10.1016/j.aim.2009.07.017.  Google Scholar

[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.  doi: 10.1007/s00205-014-0749-6.  Google Scholar

[15]

T. Kato, Perturbation theory for linear operators,, Classics in Mathematics, (1995).   Google Scholar

[16]

Y. Long, Index Theory For Symplectic Paths With Applications,, Birkhäuser Verlag, (2002).  doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

C. Zhu, A generalized Morse index theorem,, In Analysis, (2006), 493.   Google Scholar

[24]

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

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