# American Institute of Mathematical Sciences

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

## Nonlinear stability of stationary points in the problem of Robe

 1 Department Computer Science, University of Cincinnati, Cincinnati, Ohio 45221-0025, United States 2 Departamento de Matemática, Universidade Federal de Sergipe, São Cristovão-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.
Citation: Dieter Schmidt, Lucas Valeriano. Nonlinear stability of stationary points in the problem of Robe. Discrete and 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), 229-242. doi: 10.1007/BF00054966. [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), 17-30. [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), 145-155. doi: 10.1023/A:1011173320720. [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, Moscow, Izhevsk, 2009. [5] K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer, 2nd edition, 2009. [6] K. R. Meyer, J. Palacián and P. Yanguas, Stability of a Hamiltonian system in a limiting case, J. Appl. Math. Mech., 41 (2012), 20-28. [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), 222-236. doi: 10.1016/0022-0396(86)90098-7. [8] 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. [9] H. A. G. Robe, A new kind of three body problem, Celest. Mech. & Dyn. Astr., 16 (1977), 197-206. [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), 113-122. doi: 10.4236/ijaa.2013.32013. [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), 20-28. [12] L. R. Valeriano, Parametric stability in Robe's problem, Regular and Chaotic Dynamics, 21 (2016), 126-135. doi: 10.1134/S156035471601007X.

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), 229-242. doi: 10.1007/BF00054966. [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), 17-30. [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), 145-155. doi: 10.1023/A:1011173320720. [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, Moscow, Izhevsk, 2009. [5] K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer, 2nd edition, 2009. [6] K. R. Meyer, J. Palacián and P. Yanguas, Stability of a Hamiltonian system in a limiting case, J. Appl. Math. Mech., 41 (2012), 20-28. [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), 222-236. doi: 10.1016/0022-0396(86)90098-7. [8] 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. [9] H. A. G. Robe, A new kind of three body problem, Celest. Mech. & Dyn. Astr., 16 (1977), 197-206. [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), 113-122. doi: 10.4236/ijaa.2013.32013. [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), 20-28. [12] L. R. Valeriano, Parametric stability in Robe's problem, Regular and Chaotic Dynamics, 21 (2016), 126-135. doi: 10.1134/S156035471601007X.
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