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August  2016, 21(6): 1917-1936. doi: 10.3934/dcdsb.2016029

Nonlinear stability of stationary points in the problem of Robe

Dieter Schmidt 1, and  Lucas Valeriano 2,

1. 

Department Computer Science, University of Cincinnati, Cincinnati, Ohio 45221-0025, United States
2. 

Departamento de Matemática, Universidade Federal de Sergipe, São Cristovão-SE, CEP. 49100-000, Brazil

Received  August 2015 Revised  February 2016 Published  June 2016

In 1977 Robe considered a modification of the Restricted Three Body Problem, where one of the primaries is a shell filled with an incompressible liquid. The motion of the small body of negligible mass takes place inside this sphere and is therefore affected by the buoyancy force of the liquid. We investigate the existence and stability of the equilibrium points in the planar circular problem and discuss the range of the parameters for which the problem has a physical meaning.
    Our main contribution is to establish the Lyapunov stability for the equilibrium point at the center of the shell. We achieve this by putting the Hamiltonian function of Robe's problem into its normal form and then use the theorems of Arnol'd, Markeev and Sokol'skii. Resonance cases and some exceptional cases require special treatment.
Keywords: KAM theory, buoyancy., nonlinear stability, Equilibrium points.
Mathematics Subject Classification: Primary: 70F07, 70H14; Secondary: 37N0.
Citation: Dieter Schmidt, Lucas Valeriano. Nonlinear stability of stationary points in the problem of Robe. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1917-1936. doi: 10.3934/dcdsb.2016029
References:
[1]

C. M. Giordani, A. R. Plastino and A. Plastino, Robe's restricted three body-problem with drag,, Celest. Mech. & Dyn. Astr., 66 (1996), 229.  doi: 10.1007/BF00054966.  Google Scholar

[2]

P. P. Hallan and K. B. Mangang, Non linear stability of equilibrium point in the Robe's restricted circular three body problem,, Indian J. pure. appl. Math., 38 (2007), 17.   Google Scholar

[3]

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

[4]

A. P. Markeev, Linear Hamiltonian Systems and Some Applications to the Problem of Stability of Motion of Satellites Relative to the Center of Mass,, R&C Dynamics, (2009).   Google Scholar

[5]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,, Springer, (2009).   Google Scholar

[6]

K. R. Meyer, J. Palacián and P. Yanguas, Stability of a Hamiltonian system in a limiting case,, J. Appl. Math. Mech., 41 (2012), 20.   Google Scholar

[7]

K. R. Meyer and D. S. Schmidt, The stability of the Lagrange triangular point and a theorem of Arnol'd,, Journal of Differential equations, 62 (1986), 222.  doi: 10.1016/0022-0396(86)90098-7.  Google Scholar

[8]

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

[9]

H. A. G. Robe, A new kind of three body problem,, Celest. Mech. & Dyn. Astr., 16 (1977), 197.   Google Scholar

[10]

J. Singh and O. Leke, Existence and stability of equilibrium points in the Robe's restricted three-body problem with variable masses,, International Journal of Astronomy and Astrophysics, 3 (2013), 113.  doi: 10.4236/ijaa.2013.32013.  Google Scholar

[11]

A. G. Sokol'skii, On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance,, J. Appl. Math. Mech., 41 (1977), 20.   Google Scholar

[12]

L. R. Valeriano, Parametric stability in Robe's problem,, Regular and Chaotic Dynamics, 21 (2016), 126.  doi: 10.1134/S156035471601007X.  Google Scholar

show all references

References:
[1]

C. M. Giordani, A. R. Plastino and A. Plastino, Robe's restricted three body-problem with drag,, Celest. Mech. & Dyn. Astr., 66 (1996), 229.  doi: 10.1007/BF00054966.  Google Scholar

[2]

P. P. Hallan and K. B. Mangang, Non linear stability of equilibrium point in the Robe's restricted circular three body problem,, Indian J. pure. appl. Math., 38 (2007), 17.   Google Scholar

[3]

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

[4]

A. P. Markeev, Linear Hamiltonian Systems and Some Applications to the Problem of Stability of Motion of Satellites Relative to the Center of Mass,, R&C Dynamics, (2009).   Google Scholar

[5]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,, Springer, (2009).   Google Scholar

[6]

K. R. Meyer, J. Palacián and P. Yanguas, Stability of a Hamiltonian system in a limiting case,, J. Appl. Math. Mech., 41 (2012), 20.   Google Scholar

[7]

K. R. Meyer and D. S. Schmidt, The stability of the Lagrange triangular point and a theorem of Arnol'd,, Journal of Differential equations, 62 (1986), 222.  doi: 10.1016/0022-0396(86)90098-7.  Google Scholar

[8]

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

[9]

H. A. G. Robe, A new kind of three body problem,, Celest. Mech. & Dyn. Astr., 16 (1977), 197.   Google Scholar

[10]

J. Singh and O. Leke, Existence and stability of equilibrium points in the Robe's restricted three-body problem with variable masses,, International Journal of Astronomy and Astrophysics, 3 (2013), 113.  doi: 10.4236/ijaa.2013.32013.  Google Scholar

[11]

A. G. Sokol'skii, On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance,, J. Appl. Math. Mech., 41 (1977), 20.   Google Scholar

[12]

L. R. Valeriano, Parametric stability in Robe's problem,, Regular and Chaotic Dynamics, 21 (2016), 126.  doi: 10.1134/S156035471601007X.  Google Scholar

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