On stacked central configurations of the planar coorbital satellites problem

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October 2012, 32(10): 3715-3732. doi: 10.3934/dcds.2012.32.3715

On stacked central configurations of the planar coorbital satellites problem

Allyson Oliveira ^1, and Hildeberto Cabral ^2,

1.	Núcleo de Formação Docente, Universidade Federal de Pernambuco, Caruaru-PE, CEP 55002-970, Brazil
2.	Departamento de Matemática, Universidade Federal de Pernambuco, Recife-PE, CEP. 50540-740, Brazil

Received March 2011 Revised January 2012 Published May 2012

Abstract
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In this work we look for central configurations of the planar $1+n$ body problem such that, after the addition of one or two satellites, we have a new planar central configuration. We determine all such configurations in two cases: the first, the addition of two satellites considering that all satellites have equal infinitesimal masses and the second case where one satellite is added but the infinitesimal masses are not necessarily equal.

Keywords: stacked central configuration, Planar central configuration, planar coorbital satellites problem..

Mathematics Subject Classification: Primary: 70F15, 70F1.

Citation: Allyson Oliveira, Hildeberto Cabral. On stacked central configurations of the planar coorbital satellites problem. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3715-3732. doi: 10.3934/dcds.2012.32.3715

References:

[1]	A. Albouy and Y. Fu, Relative equilibria of four identical satellites, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 2633-2645. doi: 10.1098/rspa.2009.0115.
[2]	J. Casasayas, J. Llibre and A. Nunes, Central configurations of the planar $1+n$ body problem, Celestial Mech. Dynam. Astronom., 60 (1994), 273-288. doi: 10.1007/BF00693325.
[3]	M. Corbera, J. Cors and J. Llibre, On the central configurations of the planar $1+3$ body problem, Celestial Mech. Dynam. Astronom., 109 (2011), 27-43. doi: 10.1007/s10569-010-9316-0.
[4]	J. Cors, J. Llibre and M. Ollé, Central configurations of the planar coorbital satellite problem, Celestial Mech. Dynam. Astronom., 89 (2004), 319-342. doi: 10.1023/B:CELE.0000043569.25307.ab.
[5]	Y. Hagihara, "Celestial Mechanics. Volume I: Dynamical Principles and Transformation Theory," The MIT Press, Cambridge, Mass.-London, 1970.
[6]	G. Hall, Central configuration in the planar $1 + n$ body problem, preprint, 1988.
[7]	M. Hampton, Stacked central configurations: New examples in the planar five-body problem, Nonlinearity, 18 (2005), 2299-2304. doi: 10.1088/0951-7715/18/5/021.
[8]	J. Llibre and L. Mello, New central configurations for the planar 5-body problem, Celestial Mech. Dynam. Astronom., 100 (2008), 141-149. doi: 10.1007/s10569-007-9107-4.
[9]	J. Maxwell, "On the Stability of Motion of Saturn's Rings," Macmillan & Co., London, 1985.
[10]	S. Renner and B. Sicardy, Stationary configurations for co-orbital satellites with small arbitrary masses, Celestial Mech. Dynam. Astronom., 88 (2004), 397-414. doi: 10.1023/B:CELE.0000023420.80881.67.
[11]	D. Saari, On the role and the properties of $n$-body central configurations, Celestial Mech., 21 (1980), 9-20. doi: 10.1007/BF01230241.
[12]	D. Saari, "Collisions, Rings, and Other Newtonian $N$-Body Problems," CBMS Regional Conference Series in Mathematics, 104, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2005.
[13]	A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton Mathematical Series, 5, Princeton University Press, Princeton, NJ, 1941.

show all references

References:

[1]	A. Albouy and Y. Fu, Relative equilibria of four identical satellites, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 2633-2645. doi: 10.1098/rspa.2009.0115.
[2]	J. Casasayas, J. Llibre and A. Nunes, Central configurations of the planar $1+n$ body problem, Celestial Mech. Dynam. Astronom., 60 (1994), 273-288. doi: 10.1007/BF00693325.
[3]	M. Corbera, J. Cors and J. Llibre, On the central configurations of the planar $1+3$ body problem, Celestial Mech. Dynam. Astronom., 109 (2011), 27-43. doi: 10.1007/s10569-010-9316-0.
[4]	J. Cors, J. Llibre and M. Ollé, Central configurations of the planar coorbital satellite problem, Celestial Mech. Dynam. Astronom., 89 (2004), 319-342. doi: 10.1023/B:CELE.0000043569.25307.ab.
[5]	Y. Hagihara, "Celestial Mechanics. Volume I: Dynamical Principles and Transformation Theory," The MIT Press, Cambridge, Mass.-London, 1970.
[6]	G. Hall, Central configuration in the planar $1 + n$ body problem, preprint, 1988.
[7]	M. Hampton, Stacked central configurations: New examples in the planar five-body problem, Nonlinearity, 18 (2005), 2299-2304. doi: 10.1088/0951-7715/18/5/021.
[8]	J. Llibre and L. Mello, New central configurations for the planar 5-body problem, Celestial Mech. Dynam. Astronom., 100 (2008), 141-149. doi: 10.1007/s10569-007-9107-4.
[9]	J. Maxwell, "On the Stability of Motion of Saturn's Rings," Macmillan & Co., London, 1985.
[10]	S. Renner and B. Sicardy, Stationary configurations for co-orbital satellites with small arbitrary masses, Celestial Mech. Dynam. Astronom., 88 (2004), 397-414. doi: 10.1023/B:CELE.0000023420.80881.67.
[11]	D. Saari, On the role and the properties of $n$-body central configurations, Celestial Mech., 21 (1980), 9-20. doi: 10.1007/BF01230241.
[12]	D. Saari, "Collisions, Rings, and Other Newtonian $N$-Body Problems," CBMS Regional Conference Series in Mathematics, 104, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2005.
[13]	A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton Mathematical Series, 5, Princeton University Press, Princeton, NJ, 1941.

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