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Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth
Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem
1. | Department of Mathematics, Brigham Young University, Provo, UT 84602, USA |
2. | School of Mathematics and System Sciences, Beihang University, Beijing 100191, China |
We prove new variational properties of the spatial isosceles orbits in the equal-mass three-body problem and analyze their linear stabilities in both the full phase space $\mathbb{R}^{12}$ and a symmetric subspace Γ. We prove that each spatial isosceles orbit is an action minimizer of a two-point free boundary value problem with non-symmetric boundary settings. The spatial isosceles orbits form a one-parameter set with rotation angle θ as the parameter. This set of orbits always lies in a symmetric subspace Γ and we show that their linear stabilities in the full phase space $\mathbb{R}^{12}$ can be simplified to two separated sub-problems: linear stabilities in Γ and $(\mathbb{R}^{12} \setminus Γ) \cup \{0\}$. By applying Roberts' symmetry reduction method, we prove that the orbits are always unstable in the full phase space $\mathbb{R}^{12}$, but it is linearly stable in Γ when $θ ∈ [0.33π, 0.48 π] \cup [0.52 π, 0.78 π]$.
References:
[1] |
R. Broucke, On the isosceles triangle configuration in the planar general three-body problem, Astron. Astrophys., 73 (1979), 303-313. Google Scholar |
[2] |
K. 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. |
[3] |
K. Chen,
Removing collision singularities from action minimizers for the N-body problem with free boundaries, Arch. Ration. Mech. Anal., 181 (2006), 311-331.
doi: 10.1007/s00205-005-0413-2. |
[4] |
K. 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. |
[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, in Proceedings of the International Congress of Mathematiicans (Beijing, 2002), Higher Ed. Press, Beijing, 3 (2002), 279–294. |
[7] |
D. Ferrario and S. Terracini,
On the existence of collisionless equivalent minimizers for the classical n-body problem, Inv. Math., 155 (2004), 305-362.
doi: 10.1007/s00222-003-0322-7. |
[8] |
W. Gordon,
A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971.
doi: 10.2307/2373993. |
[9] |
W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, arXiv: 1607.00580. Google Scholar |
[10] |
Y. Long,
Index Theory for Symplectic Paths with Applications
Birkhäuser Verlag, Basel-Boston-Berlin, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[11] |
C. Marchal,
How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astr., 83 (2002), 325-353.
doi: 10.1023/A:1020128408706. |
[12] |
D. Offin and H. Cabral,
Hyperbolicity for symmetric periodic orbits in the isosceles three body problem, Dis. Con. Dyn. Syst. Ser. S, 2 (2009), 379-392.
doi: 10.3934/dcdss.2009.2.379. |
[13] |
G. Roberts,
Linear Stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dyn. Sys., 27 (2007), 1947-1963.
doi: 10.1017/S0143385707000284. |
[14] |
M. Shibayama,
Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155.
|
[15] |
D. Yan,
Existence of the Broucke periodic orbit and its linear stability, J. Math. Anal. Appl., 389 (2012), 656-664.
doi: 10.1016/j.jmaa.2011.12.024. |
[16] |
D. Yan and T. Ouyang,
New phenomena in the spatial isosceles three-body problem, Int. J. Bifurcation Chaos, 25 (2015), 1550116.
doi: 10.1142/S0218127415501163. |
[17] |
D. Yan, R. Liu, X. Hu, W. Mao and T. Ouyang,
New phenomena in the spatial isosceles three-body problem with unequal masses, Int. J. Bifurcation Chaos, 25 (2015), 1550169.
doi: 10.1142/S0218127415501692. |
[18] |
Personal communications with chongchun zeng at Georgia institute of technology, 2008. Google Scholar |
show all references
References:
[1] |
R. Broucke, On the isosceles triangle configuration in the planar general three-body problem, Astron. Astrophys., 73 (1979), 303-313. Google Scholar |
[2] |
K. 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. |
[3] |
K. Chen,
Removing collision singularities from action minimizers for the N-body problem with free boundaries, Arch. Ration. Mech. Anal., 181 (2006), 311-331.
doi: 10.1007/s00205-005-0413-2. |
[4] |
K. 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. |
[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, in Proceedings of the International Congress of Mathematiicans (Beijing, 2002), Higher Ed. Press, Beijing, 3 (2002), 279–294. |
[7] |
D. Ferrario and S. Terracini,
On the existence of collisionless equivalent minimizers for the classical n-body problem, Inv. Math., 155 (2004), 305-362.
doi: 10.1007/s00222-003-0322-7. |
[8] |
W. Gordon,
A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971.
doi: 10.2307/2373993. |
[9] |
W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, arXiv: 1607.00580. Google Scholar |
[10] |
Y. Long,
Index Theory for Symplectic Paths with Applications
Birkhäuser Verlag, Basel-Boston-Berlin, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[11] |
C. Marchal,
How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astr., 83 (2002), 325-353.
doi: 10.1023/A:1020128408706. |
[12] |
D. Offin and H. Cabral,
Hyperbolicity for symmetric periodic orbits in the isosceles three body problem, Dis. Con. Dyn. Syst. Ser. S, 2 (2009), 379-392.
doi: 10.3934/dcdss.2009.2.379. |
[13] |
G. Roberts,
Linear Stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dyn. Sys., 27 (2007), 1947-1963.
doi: 10.1017/S0143385707000284. |
[14] |
M. Shibayama,
Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155.
|
[15] |
D. Yan,
Existence of the Broucke periodic orbit and its linear stability, J. Math. Anal. Appl., 389 (2012), 656-664.
doi: 10.1016/j.jmaa.2011.12.024. |
[16] |
D. Yan and T. Ouyang,
New phenomena in the spatial isosceles three-body problem, Int. J. Bifurcation Chaos, 25 (2015), 1550116.
doi: 10.1142/S0218127415501163. |
[17] |
D. Yan, R. Liu, X. Hu, W. Mao and T. Ouyang,
New phenomena in the spatial isosceles three-body problem with unequal masses, Int. J. Bifurcation Chaos, 25 (2015), 1550169.
doi: 10.1142/S0218127415501692. |
[18] |
Personal communications with chongchun zeng at Georgia institute of technology, 2008. Google Scholar |







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