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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1098/rspa.2009.0115.  Google Scholar

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Y. Hagihara, "Celestial Mechanics. Volume I: Dynamical Principles and Transformation Theory,", The MIT Press, (1970).   Google Scholar

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G. Hall, Central configuration in the planar $1 + n$ body problem,, preprint, (1988).   Google Scholar

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M. Hampton, Stacked central configurations: New examples in the planar five-body problem,, Nonlinearity, 18 (2005), 2299.  doi: 10.1088/0951-7715/18/5/021.  Google Scholar

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S. Renner and B. Sicardy, Stationary configurations for co-orbital satellites with small arbitrary masses,, Celestial Mech. Dynam. Astronom., 88 (2004), 397.  doi: 10.1023/B:CELE.0000023420.80881.67.  Google Scholar

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D. Saari, On the role and the properties of $n$-body central configurations,, Celestial Mech., 21 (1980), 9.  doi: 10.1007/BF01230241.  Google Scholar

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D. Saari, "Collisions, Rings, and Other Newtonian $N$-Body Problems,", CBMS Regional Conference Series in Mathematics, 104 (2005).   Google Scholar

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A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, 5 (1941).   Google Scholar

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.  doi: 10.1098/rspa.2009.0115.  Google Scholar

[2]

J. Casasayas, J. Llibre and A. Nunes, Central configurations of the planar $1+n$ body problem,, Celestial Mech. Dynam. Astronom., 60 (1994), 273.  doi: 10.1007/BF00693325.  Google Scholar

[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.  doi: 10.1007/s10569-010-9316-0.  Google Scholar

[4]

J. Cors, J. Llibre and M. Ollé, Central configurations of the planar coorbital satellite problem,, Celestial Mech. Dynam. Astronom., 89 (2004), 319.  doi: 10.1023/B:CELE.0000043569.25307.ab.  Google Scholar

[5]

Y. Hagihara, "Celestial Mechanics. Volume I: Dynamical Principles and Transformation Theory,", The MIT Press, (1970).   Google Scholar

[6]

G. Hall, Central configuration in the planar $1 + n$ body problem,, preprint, (1988).   Google Scholar

[7]

M. Hampton, Stacked central configurations: New examples in the planar five-body problem,, Nonlinearity, 18 (2005), 2299.  doi: 10.1088/0951-7715/18/5/021.  Google Scholar

[8]

J. Llibre and L. Mello, New central configurations for the planar 5-body problem,, Celestial Mech. Dynam. Astronom., 100 (2008), 141.  doi: 10.1007/s10569-007-9107-4.  Google Scholar

[9]

J. Maxwell, "On the Stability of Motion of Saturn's Rings,", Macmillan & Co., (1985).   Google Scholar

[10]

S. Renner and B. Sicardy, Stationary configurations for co-orbital satellites with small arbitrary masses,, Celestial Mech. Dynam. Astronom., 88 (2004), 397.  doi: 10.1023/B:CELE.0000023420.80881.67.  Google Scholar

[11]

D. Saari, On the role and the properties of $n$-body central configurations,, Celestial Mech., 21 (1980), 9.  doi: 10.1007/BF01230241.  Google Scholar

[12]

D. Saari, "Collisions, Rings, and Other Newtonian $N$-Body Problems,", CBMS Regional Conference Series in Mathematics, 104 (2005).   Google Scholar

[13]

A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, 5 (1941).   Google Scholar

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