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March  2017, 37(3): 1763-1787. doi: 10.3934/dcds.2017074

Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

Qinglong Zhou 1,, and  Yongchao Zhang 2,

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China
2. 

School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, Hebei, China

* Corresponding author

Received  April 2016 Revised  October 2016 Published  December 2016

Fund Project: The first author is supported by NSFC (No. 11501330, No. 11425105) and CPSF (Grant No. 2015M582071). The second author is supported by the Fundamental Research Funds for the Central Universities (Grant No. N142303010).

We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in [13], which is used to model small oscillations of the earth's inner core taking into account the moon's attraction. Earlier results for the linear stability of the elliptic equilibrium point in Robe's restricted problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic equilibrium point in our problem coincides with the linearized system near the Euler elliptic relative equilibria in the classical three-body problem except for the range of the mass parameter. We first establish some relations of the linear stability problem to the properties of some symplectic paths and some corresponding linear operators. Then using the Maslov-type $ω$-index theory of symplectic paths and the theory of linear operators, we compute $ω$-indices and obtain certain properties of the linear stability of the elliptic equilibrium point of Robe's restricted three-body problem.

Keywords: Restricted three-body problem, equilibrium point, linear stability, Maslov-type $ω$-index.
Mathematics Subject Classification: Primary:70F07, 70H14;Secondary:34C25.
Citation: Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074
References:
[1]

W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics on positively curved manifolds, Ann. of Math., 116 (1982), 213-247.  doi: 10.2307/2007062.

[2]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[3]

P. P. Hallen and D. N. Rana, The Existence and stability of equilibrium points in the Robe's restricted three-body problem, Celest. Mech. Dyn. Astr., 79 (2001), 145-155.  doi: 10.1023/A:1011173320720.

[4]

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

[5]

X. Hu and Y. Ou, Collision index and stability of elliptic relative equilibria in planar $n$-body problem, Commun. Math. Phys., 348 (2016), 803-845.  doi: 10.1007/s00220-016-2695-7.

[6]

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

[7]

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.

[8]

Y. Long, The structure of the singular symplectic matrix set, Sci. China. Ser. A. (English Ed.), 34 (1991), 897-907. 

[9]

Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.

[10]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914.

[11]

Y. Long, Index Theory for Symplectic Paths with Applications Progress in Math. 207, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.

[12]

A. R. Plastino and A. Plastino, Robe's restricted three-body problem revisited, Celest. Mech. Dyn. Astr., 61 (1995), 197-206.  doi: 10.1007/BF00048515.

[13]

H. A. G. Robe, A new kind of three-body problem, Celest. Mech., 16 (1977), 343-351.  doi: 10.1007/BF01232659.

[14]

A. K. Shrivastava and D. Garain, Effect of perturbation on the location of liberation point in the Robe restricted problem of three bodies, Celest. Mech., 51 (1991), 67-73.  doi: 10.1007/BF02426670.

[15]

K. T. Singh, B. S. Kushvah and B. Ishwar, Stability of triangular equilibrium points in Robe's generalised restricted three body problem, in Celestial Mechanics: Recent Trends (eds. B.Ishwar), Narosa Publishing House Pvt. Ltd., New Delhi, India, (2006), 65-70. 

[16]

Q. Zhou and Y. Long, Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems, J. Diff. Equa., 258 (2015), 3851-3879.  doi: 10.1016/j.jde.2015.01.045.

[17]

Q. Zhou and Y. Long, Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem, preprint, arXiv: 1510.06822.

[18]

Q. Zhou and Y. Long, The reduction on the linear stability of elliptic Euler-Moulton solutions of the $n$-body problem to those of $3$-body problems, Celest. Mech. Dyn. Astr., (2016), 1-32.  doi: 10.1007/s10569-016-9732-x.

show all references

References:
[1]

W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics on positively curved manifolds, Ann. of Math., 116 (1982), 213-247.  doi: 10.2307/2007062.

[2]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[3]

P. P. Hallen and D. N. Rana, The Existence and stability of equilibrium points in the Robe's restricted three-body problem, Celest. Mech. Dyn. Astr., 79 (2001), 145-155.  doi: 10.1023/A:1011173320720.

[4]

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

[5]

X. Hu and Y. Ou, Collision index and stability of elliptic relative equilibria in planar $n$-body problem, Commun. Math. Phys., 348 (2016), 803-845.  doi: 10.1007/s00220-016-2695-7.

[6]

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

[7]

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.

[8]

Y. Long, The structure of the singular symplectic matrix set, Sci. China. Ser. A. (English Ed.), 34 (1991), 897-907. 

[9]

Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.

[10]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914.

[11]

Y. Long, Index Theory for Symplectic Paths with Applications Progress in Math. 207, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.

[12]

A. R. Plastino and A. Plastino, Robe's restricted three-body problem revisited, Celest. Mech. Dyn. Astr., 61 (1995), 197-206.  doi: 10.1007/BF00048515.

[13]

H. A. G. Robe, A new kind of three-body problem, Celest. Mech., 16 (1977), 343-351.  doi: 10.1007/BF01232659.

[14]

A. K. Shrivastava and D. Garain, Effect of perturbation on the location of liberation point in the Robe restricted problem of three bodies, Celest. Mech., 51 (1991), 67-73.  doi: 10.1007/BF02426670.

[15]

K. T. Singh, B. S. Kushvah and B. Ishwar, Stability of triangular equilibrium points in Robe's generalised restricted three body problem, in Celestial Mechanics: Recent Trends (eds. B.Ishwar), Narosa Publishing House Pvt. Ltd., New Delhi, India, (2006), 65-70. 

[16]

Q. Zhou and Y. Long, Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems, J. Diff. Equa., 258 (2015), 3851-3879.  doi: 10.1016/j.jde.2015.01.045.

[17]

Q. Zhou and Y. Long, Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem, preprint, arXiv: 1510.06822.

[18]

Q. Zhou and Y. Long, The reduction on the linear stability of elliptic Euler-Moulton solutions of the $n$-body problem to those of $3$-body problems, Celest. Mech. Dyn. Astr., (2016), 1-32.  doi: 10.1007/s10569-016-9732-x.

Figure 1.  The Robe's restricted three-body problem considered: $m_1$ is a spherical shell filled with a fluid of density $\rho_1$; $m_2$ a mass point outside the shell and $m_3$ a small solid sphere of density $\rho_3$ inside the shell.
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Figure 2.  Stability bifurcation diagram of elliptic equilibrium point of the Robe's restricted three-body problem in the $(\mu,e)$ rectangle $[0,1]\times [0,1)$.
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