Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 272791, 5 pages http://dx.doi.org/10.1155/2013/272791 Research Article Central Configurations for Newtonian + 2 + 1-Body Problems Furong Zhao 1,2 and Jian Chen 1,3 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 Department of Mathematics and Computer Science, Mianyang Normal University, Mianyang, Sichuan 621000, China 3 Department of Mathematics, Southwest University of Science and Technology, Mianyang, Sichuan 621000, China Correspondence should be addressed to Furong Zhao; [email protected] Received 29 November 2012; Revised 31 January 2013; Accepted 3 February 2013 Academic Editor: Baodong Zheng Copyright © 2013 F. Zhao and J. Chen. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We show the existence of spatial central configurations for the + 2 + 1-body problems. In the + 2 + 1-body problems, N bodies are at the vertices of a regular N-gon T ; 2p bodies are symmetric with respect to the center of T, and located on the straight line which is perpendicular to the regular N-gon T and passes through the center of T; the +2+1th is located at the the center of T. e masses located on the vertices of the regular N-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center of T are equal. 1. Introduction and Main Results e Newtonian -body problems [1–3] concern with the motions of particles with masses ∈ + and positions ∈ 3 ( = 1,2,...,), and the motion is governed by Newton’s second law and the Universal law: .. = () , (1) where = ( 1 , 2 ,..., ) and with Newtonian potential: () = ∑ 1⩽<⩽ − . (2) Consider the space = { { { = ( 1 , 2 ,..., )∈ 3 : ∑ =1 =0 } } } , (3) that is, suppose that the center of mass is fixed at the origin of the space. Because the potential is singular when two particles have the same position, it is natural to assume that the configuration avoids the collision set Δ = { = ( 1 ,..., ): = for some / = }. e set \Δ is called the configuration space. Definition 1 (see [2, 3]). A configuration = ( 1 , 2 ,..., )∈ \Δ is called a central configuration if there exists a constant such that ∑ =1, / = − 3 ( − ) = − , 1 ⩽ ⩽ . (4) e value of constant in (4) is uniquely determined by = , (5) where = ∑ =1 2 . (6) Since the general solution of the -body problem cannot be given, great importance has been attached to search for particular solutions from the very beginning. A homographic solution is a configuration which is preserved for all time. Central configurations and homographic solutions are linked by the Laplace theorem [3]. Collapse orbits and parabolic orbits have relations with the central configurations [2, 4– 6], so finding central configurations becomes very important. e main general open problem for the central configurations
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 272791 5 pageshttpdxdoiorg1011552013272791
Research ArticleCentral Configurations for Newtonian119873+ 2119901 + 1-Body Problems
Furong Zhao12 and Jian Chen13
1 Department of Mathematics Sichuan University Chengdu Sichuan 610064 China2Department of Mathematics and Computer Science Mianyang Normal University Mianyang Sichuan 621000 China3Department of Mathematics Southwest University of Science and Technology Mianyang Sichuan 621000 China
Correspondence should be addressed to Furong Zhao zhaofurong2006163com
Received 29 November 2012 Revised 31 January 2013 Accepted 3 February 2013
Academic Editor Baodong Zheng
Copyright copy 2013 F Zhao and J ChenThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We show the existence of spatial central configurations for the 119873 + 2119901 + 1-body problems In the 119873 + 2119901 + 1-body problems Nbodies are at the vertices of a regular N-gon T 2p bodies are symmetric with respect to the center of T and located on the straightline which is perpendicular to the regular N-gon T and passes through the center of T the119873+2119901+ 1th is located at the the centerof T The masses located on the vertices of the regular N-gon are assumed to be equal the masses located on the same line andsymmetric with respect to the center of T are equal
1 Introduction and Main Results
The Newtonian 119899-body problems [1ndash3] concern with themotions of 119899 particles with masses 119898119895 isin 119877+ and positions119902119895 isin 1198773 (119895 = 1 2 119899) and the motion is governed byNewtonrsquos second law and the Universal law119898119895 119902119895 = 120597119880 (119902)120597119902119895 (1)
where 119902 = (1199021 1199022 119902119899) and with Newtonian potential119880 (119902) = sum1⩽119895lt119896⩽119899
11989811989511989811989610038161003816100381610038161003816119902119895 minus 11990211989610038161003816100381610038161003816 (2)
Consider the space
119883 = 119902 = (1199021 1199022 119902119899) isin 1198773119899 119873sum119895=1
119898119895119902119895 = 0 (3)
that is suppose that the center of mass is fixed at the originof the space Because the potential is singular when twoparticles have the same position it is natural to assumethat the configuration avoids the collision set Δ = 119902 =(1199021 119902119873) 119902119895 = 119902119896 for some 119896 = 119895 The set 119883 Δ is calledthe configuration space
Definition 1 (see [2 3]) A configuration 119902 = (1199021 1199022 119902119899) isin119883Δ is called a central configuration if there exists a constant120582 such that119899sum119895=1119895 =119896
11989811989511989811989610038161003816100381610038161003816119902119895 minus 119902119896100381610038161003816100381610038163 (119902119895 minus 119902119896) = minus120582119898119896119902119896 1 ⩽ 119896 ⩽ 119899 (4)
The value of constant 120582 in (4) is uniquely determined by120582 = 119880119868 (5)
where 119868 = 119899sum119896=1
119898119896100381610038161003816100381611990211989610038161003816100381610038162 (6)
Since the general solution of the 119899-body problem cannotbe given great importance has been attached to search forparticular solutions from the very beginning A homographicsolution is a configuration which is preserved for all timeCentral configurations and homographic solutions are linkedby the Laplace theorem [3] Collapse orbits and parabolicorbits have relations with the central configurations [2 4ndash6] so finding central configurations becomes very importantThemain general open problem for the central configurations
2 Abstract and Applied Analysis
119910
1198981
1198982
1198983
1198984
1198985 119898611989871198988
1198989
119909
119911
Figure 1
is due to Wintner [3] and Smale [7] is the number ofcentral configurations finite for any choice of positive masses1198981 119898119899 Hampton and Moeckel [8] have proved thisconjecture for any given four positive masses
For 5-body problems Hampton [9] provided a newfamily of planar central configurations called stacked centralconfigurations A stacked central configuration is one thathas some proper subset of three or more points forming acentral configuration Ouyang et al [10] studied pyramidalcentral configurations for Newtonian119873 + 1-body problemsZhang and Zhou [11] considered double pyramidal centralconfigurations for Newtonian 119873 + 2-body problems Melloand Fernandes [12] analyzed new classes of spatial centralconfigurations for the119873+ 3-body problem Llibre and Mellostudied triple and quadruple nested central configurations forthe planar 119899-body problem There are many papers studyingcentral configuration problems such as [13ndash22]
Based on the above works we study stacked centralconfiguration for Newtonian 119873 + 2119901 + 1-body problems Inthe119873+2119901+1-body problems119873 bodies are at the vertices ofa regular 119873-gon 119879 and 2119901 bodies are symmetrically locatedon the same straight line which is perpendicular to 119879 andpasses through the center of 119879 the 119873 + 2119901 + 1th body islocated at the center of 119879 The masses located on the verticesof the regular119873-gon are equal the masses located on the lineand symmetric with respect to the center of 119879 are equal (seeFigure 1 for119873 = 4 and 119901 = 2)
In this paper we will prove the following result
Theorem 2 For119873+ 2119901 + 1-body problem in 1198773 where119873 ge 2and 119901 ge 1 there is at least one central configuration such that119873 bodies are at the vertices of a regular119873-gon119879 and 2119901 bodiesare symmetric with respect to the center of the regular 119873-gon119879 and located on a line which is perpendicular to the regular119873-gon119879 the119873+2119901+1th body is located at the center of119879Themasses at the vertices of 119879 are equal and the masses symmetricwith respect to the center of 119879 are equal
2 The Proof of Theorem 2
Our approach to Theorem 2 is inspired by the of argumentsof Corbera et al in [23]
21 Equations for the Central Configurations of 119873 + 2119901-Body Problems To begin we take a coordinate system whichsimplifies the analysis The particles have positions given by119902119895 = (cos120572119895 sin120572119895 0) where 120572119895 = ((119895 minus 1)119873)2120587 119895 =1 119873 119902119873+119895 = (0 0 119903119895) 119902119873+119895+119901 = (0 0 minus119903119895) where 119895 =1 119901 119902119873+2119901+1 = (0 0 0)
The masses are given by 1198981 = 1198982 = sdot sdot sdot = 119898119873 = 1119898119873+119895 = 119898119873+119895+119901 = 119872119895 where 119895 = 1 119901 119898119873+2119901+1 = 1198720Notice that (1199021 119902119873 119902119873+1 119902119873+2119901 119902119873+2119901+1) is a cen-
tral configuration if and only if
119873+2119901+1sum119895=1119895 =119896
11989811989511989811989610038161003816100381610038161003816119902119895 minus 119902119896100381610038161003816100381610038163 (119902119895 minus 119902119896) = minus120582119898119896119902119896 1 ⩽ 119896 ⩽ 119873 + 2119901(7)
By the symmetries of the system (7) is equivalent to thefollowing equations
119873+2119901+1sum119895=1119895 =119896
11989811989511989811989610038161003816100381610038161003816119902119895 minus 119902119896100381610038161003816100381610038163 (119902119895 minus 119902119896) = minus120582119898119896119902119896119896 = 1119873 + 1 119873 + 119901 (8)
that is minus120582 (1 0 0) = minus 120573 (1 0 0) minus (1 0 0)1198720minus 119901sum119894=1
210038161003816100381610038161 + 1199032119894 100381610038161003816100381632119872119894 (1 0 0) (9)
where 120573 = (14)sum119873minus1119895=1 csc(120587119895119873)minus120582 (0 0 1199031) = minus 11987310038161003816100381610038161 + 11990321 100381610038161003816100381632 (0 0 1199031) minus (0 0 111990321 )1198720minus 1100381610038161003816100381621199031100381610038161003816100381621198721 (0 0 1)+ 119901sum119894=2
[ 11003816100381610038161003816119903119894 minus 119903110038161003816100381610038162 minus 11003816100381610038161003816119903119894 + 119903110038161003816100381610038162]1198721 (0 0 1) minus120582 (0 0 119903119895) = minus 119873100381610038161003816100381610038161 + 1199032119895 1003816100381610038161003816100381632 (0 0 119903119895) minus (0 0 11199032119895 )1198720minus sum1le119894lt119895
( 110038161003816100381610038161003816119903119894 minus 119903119895100381610038161003816100381610038162 + 110038161003816100381610038161003816119903119894 + 119903119895100381610038161003816100381610038162)119872119894 (0 0 1)minus 1100381610038161003816100381610038162119903119895100381610038161003816100381610038162119872119895 (0 0 1)+ sum119895lt119894le119901
( 110038161003816100381610038161003816119903119894 minus 119903119895100381610038161003816100381610038162 minus 110038161003816100381610038161003816119903119894 + 119903119895100381610038161003816100381610038162)119872119894 (0 0 1) 119895 = 2 119901 minus 1(10)
Abstract and Applied Analysis 3
minus120582 (0 0 119903119901) = minus 119873100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381632 (0 0 119903119901) minus (0 0 11199032119901)1198720minus sum1le119894lt119901
( 110038161003816100381610038161003816119903119894 minus 119903119901100381610038161003816100381610038162 + 110038161003816100381610038161003816119903119894 + 119903119901100381610038161003816100381610038162)119872119894 (0 0 1)minus 1100381610038161003816100381610038162119903119901100381610038161003816100381610038162119872119901 (0 0 1)
(11)
In order to simplify the equations we defined1198860119895 = minus2|1+1199032119895 |32 119886119895119895 = minus141199033119895 where 119895 = 1 119901119886119894119895 = (1|119903119894 minus 119903119895|2119903119894 minus 1|119903119894 + 119903119895|2119903119894) when 119894 lt 119895119886119894119895 = minus(1|119903119894 minus 119903119895|2119903119894 + 1|119903119894 + 119903119895|2119903119894) when 119894 gt 11989511988600 = 1198861198940 = 1 when 119894 = 1 1199011198870 = 120573 + 1198720 119887119894 = 119873|1 + 1199032119894 |32 + 11987201199033119894 where 119894 =1 119901Equations (9)ndash(11) can be written as a linear system of theform 119860119883 = 119887 given by
(1 11988601 11988602 11988603 sdot sdot sdot 11988601199011 11988611 11988612 11988613 sdot sdot sdot 11988611199011 11988621 11988622 11988623 sdot sdot sdot 1198862119901
1 1198861199011 1198861199012 1198861199013 sdot sdot sdot 119886119901119901)(((
120582119872111987221198723119872119875)))
=(((
1198870119887111988721198873119887119901)))
(12)
The column vector is given by the variables 119883 = (12058211987211198722 119872119901)119879 Since the 119886119894119895 is function of 1199031 1199032 119903119901 wewrite the coefficient matrix as 119860119901(1199031 1199032 119903119901)22 For 119901 = 1 We need the next lemma
Lemma 3 (see [12]) Assuming 1198981 = sdot sdot sdot = 119898119873 =1 119898119873+1 = 119898119873+2 = 1198721 there is a nonempty interval119868 sub 119877 1198720(1199031) and 1198721(1199031) such that for each 1199031 isin 119868(1199021 119902119873 119902119873+1 119902119873+2 119902119873+3) forms a central configuration ofthe119873 + 2 + 1-body problem
For 119901 = 1 system (12) becomes(1 119886011 11988611)( 1205821198721) = (11988701198871) (13)10038161003816100381610038161198601 (1199031)1003816100381610038161003816 = 11988611 minus 11988601 = minus 1411990331 + 210038161003816100381610038161 + 11990321 100381610038161003816100381632 (14)
If we consider |1198601(1199031)| as a function of 1199031 then |1198601(1199031)| is ananalytic function and nonconstant By Lemma 3 there exists
a r1 isin 119868 such that (13) has a unique solution (1205821198721) satisfying120582 gt 0 and1198721 gt 023 For All 119901gt1 The proof for 119901 ge 1 is done by inductionWe claim that there exists 0 lt 1199031 lt 1199032 lt sdot sdot sdot lt 119903119901 such thatsystem (12) has a unique solution 120582 = 120582(1199031 119903119901) gt 0119872119894 =119872119894(1199031 119903119901) gt 0 for 119894 = 1 119901 We have seen that theclaim is true for 119901 = 1 We assume the claim is true for 119901 minus 1and we will prove it for 119901 Assume by induction hypothesisthat there exists 0 lt 1 lt 2 lt sdot sdot sdot lt 119901minus1 such that system(12) has a unique solution = 120582(1 2 119901minus1) gt 0 and119894 = 119872119894(1 2 119901minus1) gt 0 for 119894 = 1 2 119901 minus 1
We need the next lemma
Lemma 4 There exists 119901 gt 119901minus1 such that = 120582(1 2 119901minus1) 119894 = 119872119894(1 2 119901minus1) for 119894 = 1 2 119901 minus 1 and119901 = 0 is a solution of (12)
Proof Since 119872119901 = 119901 = 0 we have that the first 119901 minus 1equation of (12) is satisfied when 120582 = 119872119894 = 119894 for 119894 =1 2 119901 minus 1 and119872119901 = 119901 = 0 Substituting this solutioninto the last equation of (12) we let119891 (119903119901) = + 11988611990111 + sdot sdot sdot + 119886119901119901minus1119901minus1 minus 119873100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381632 minus 11987201199033119901
(15)
We have that
lim119903119901rarr+infin
119891 (119903119901) = gt 0 lim119903119901rarr
+119901minus1
119891 (119903119901) = minusinfin (16)
Therefore there exists at least a value 119903119901 = 119901 gt 119901minus1 satisfyingequation 119891(119903119901) = 0 This completes the proof of Lemma 4
By using the implicit function theorem we will prove thatthe solution of (12) given in Lemma 3 can be continued to asolution with119872119901 gt 0
Let 119904 = (120582 1199031 1199031199011198721 119872119901) we define1198920 (119904) = 120582 + 119886011198721 + sdot sdot sdot + 1198860119901119872119901 minus 11988701198921 (119904) = 120582 + 119886111198721 + sdot sdot sdot + 1198861119901119872119901 minus 11988711198922 (119904) = 120582 + 119886211198721 + sdot sdot sdot + 1198862119901119872119901 minus 1198872119892119901 (119904) = 120582 + 11988611990111198721 + sdot sdot sdot + 119886119901119901119872119901 minus 119887119901
(17)
It is not difficult to see that the system (12) is equivalent to119892119894(119904) = 0 for 119894 = 0 1 119901Let = ( 1 2 119901 1 119901) be the solution of
system (12) given in Lemma 4 The differential of (17) with
4 Abstract and Applied Analysis
respect to the variables (12058211987211198722 119872119901minus1 119903119901) is119863119901 () =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 11988601 11988602 sdot sdot sdot 1198860119901minus1 01 11988611 11988612 sdot sdot sdot 1198861119901minus1 0
1 119886119901minus11 119886119901minus12 sdot sdot sdot 119886119901minus1119901minus1 01 1198861199011 1198861199012 sdot sdot sdot 119886119901119901minus1 120597119892119901 ()1205971199031199011003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816119860119901minus1 (1 2 119901minus1)10038161003816100381610038161003816 120597119892119901 ()120597119903119901 120597119892119901 ()120597119903119901 = 119901minus1sum
119894=1
[[ 210038161003816100381610038161003816119903119901 minus 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 minus 1199031198941003816100381610038161003816100381621199032119901+ 210038161003816100381610038161003816119903119901 + 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 + 1199031198941003816100381610038161003816100381631199032119901]]119894+ (3119873119903119901)100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381652 + 311987201199034119901 gt 0
(18)
Wehave assumed that 0 lt 1 lt 2 lt sdot sdot sdot lt 119901minus1exists such thatsystem (12) with 119901 minus 1 instead of 119901 has a unique solution so|119860119901minus1(1 2 119901minus1)| = 0 therefore 119863() = 0 Applying theImplicit Function Theorem there exists a neighborhood 119880of (119901 1 1 119901minus1) and unique analytic functions 120582 =120582(119901 1 1 119901minus1) 119872119894 = 119872119894(119901 1 1 119901minus1) for 119894 =1 119901 minus 1 and 119903119901 = 119903119901(119872119901 1199031 1199032 119903119901minus1) such that(12058211987211198722 119872119901) is the solution of the system (12) for all(119872119901 1199031 1199032 119903119901minus1) isin 119880 The determinant is calculated as10038161003816100381610038161003816119860119901 (1199031 1199032 119903119901)10038161003816100381610038161003816= 119886119901119901 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum
119894=0
119886119901119894119861119901119894= minus1100381610038161003816100381610038162119903119901100381610038161003816100381610038162 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum119894=0119886119901119894119861119901119894
(19)
where 119861119901119894 is the algebraic cofactor of 119886119901119894We see that |119860119901minus1(1199031 1199032 119903119901minus1)| 119886119901119894 and 119861119901119894 for119894 = 0 1 119901 minus 1 do not contain the factor 1|2119903119901|2
If we consider |119860119901(1199031 1199032 119903119901)| as a function of 119903119901then |119860119901(1199031 1199032 119903119901)| is analytic and nonconstant Wecan find (1199031 1199032 119903119901) sufficiently close to (1 2 119901)such that |119860119901(1199031 1199032 119903119901)| = 0 and therefore a solution(1205821198721 119872119901) of system (12) is satisfying 120582 gt 0 119872119894 gt 0for 119894 = 1 119901
The proof of Theorem 2 is completed
Acknowledgments
The authors express their gratitude to Professor ZhangShiqing for his discussions and helpful suggestionsThiswork
is supported by NSF of China and Youth found of MianyangNormal University
References
[1] R Abraham and J E Marsden Foundation of MechanicsBenjamin New York NY USA 2nd edition 1978
[2] D G Saari Rings and Other Newtonian N-body ProblemsAmerican Mathematical Society Providence RI USA 2005
[3] A Wintner The Analytical Foundations of Celestial Mechanicsvol 5 of Princeton Mathematical Series Princeton UniversityPress Princeton NJ USA 1941
[4] D G Saari ldquoSingularities and collisions of Newtonian gravi-tational systemsrdquo Archive for Rational Mechanics and Analysisvol 49 pp 311ndash320 1973
[5] D G Saari ldquoOn the role and the properties of n-body centralconfigurationsrdquo Celestial Mechanics and Dynamical Astronomyvol 21 no 1 pp 9ndash20 1980
[6] D G Saari and N D Hulkower ldquoOn the manifolds of totalcollapse orbits and of completely parabolic orbits for the n-bodyproblemrdquo Journal of Differential Equations vol 41 no 1 pp 27ndash43 1981
[7] S Smale ldquoMathematical problems for the next centuryrdquoMath-ematical Intelligencer vol 20 pp 141ndash145 1998
[8] M Hampton and R Moeckel ldquoFiniteness of relative equilibriaof the four-body problemrdquo Inventiones Mathematicae vol 163no 2 pp 289ndash312 2006
[9] M Hampton ldquoStacked central configurations new examples inthe planar five-body problemrdquo Nonlinearity vol 18 no 5 pp2299ndash2304 2005
[10] T Ouyang Z Xie and S Zhang ldquoPyramidal central configura-tions and perverse solutionsrdquo Electronic Journal of DifferentialEquations vol 2004 no 106 pp 1ndash9 2004
[11] S Zhang and Q Zhou ldquoDouble pyramidal central configura-tionsrdquo Physics Letters A vol 281 no 4 pp 240ndash248 2001
[12] L F Mello and A C Fernandes ldquoNew classes of spatial centralconfigurations for the n-body problemrdquo Nonlinear AnalysisReal World Applications vol 12 no 1 pp 723ndash730 2011
[13] A Albouy ldquoThe symmetric central configurations of four equalmassesrdquo in Hamiltonian Dynamics and Celestial Mechanicsvol 198 of Contemporary Mathematics pp 131ndash135 AmericanMathematical Society Providence RI USA 1996
[14] A Albouy Y Fu and S Sun ldquoSymmetry of planar four-bodyconvex central configurationsrdquo Proceedings of The Royal Societyof London Series A vol 464 no 2093 pp 1355ndash1365 2008
[15] F N Diacu ldquoThe masses in a symmetric centered solution ofthe n-body problemrdquo Proceedings of the AmericanMathematicalSociety vol 109 no 4 pp 1079ndash1085 1990
[16] J Lei and M Santoprete ldquoRosette central configurationsdegenerate central configurations and bifurcationsrdquo CelestialMechanics amp Dynamical Astronomy vol 94 no 3 pp 271ndash2872006
[17] R Moeckel ldquoOn central configurationsrdquo MathematischeZeitschrift vol 205 no 4 pp 499ndash517 1990
[18] R Moeckel and C Simo ldquoBifurcation of spatial central con-figurations from planar onesrdquo SIAM Journal on MathematicalAnalysis vol 26 no 4 pp 978ndash998 1995
[19] FR Moulton ldquoThe straight line solutions of the n-body prob-lemrdquoAnnals ofMathematics Second Series vol 12 pp 1ndash17 1910
Abstract and Applied Analysis 5
[20] L M Perko and E L Walter ldquoRegular polygon solutions of theN-body problemrdquo Proceedings of the American MathematicalSociety vol 94 no 2 pp 301ndash309 1985
[21] J Shi and Z Xie ldquoClassification of four-body central configura-tions with three equal massesrdquo Journal ofMathematical Analysisand Applications vol 363 no 2 pp 512ndash524 2010
[22] Z Xia ldquoCentral configurations with many small massesrdquoJournal of Differential Equations vol 91 no 1 pp 168ndash179 1991
[23] M Corbera J Delgado and J Llibre ldquoOn the existence ofcentral configurations of p nested n-gonsrdquo Qualitative Theoryof Dynamical Systems vol 8 no 2 pp 255ndash265 2009
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
119910
1198981
1198982
1198983
1198984
1198985 119898611989871198988
1198989
119909
119911
Figure 1
is due to Wintner [3] and Smale [7] is the number ofcentral configurations finite for any choice of positive masses1198981 119898119899 Hampton and Moeckel [8] have proved thisconjecture for any given four positive masses
For 5-body problems Hampton [9] provided a newfamily of planar central configurations called stacked centralconfigurations A stacked central configuration is one thathas some proper subset of three or more points forming acentral configuration Ouyang et al [10] studied pyramidalcentral configurations for Newtonian119873 + 1-body problemsZhang and Zhou [11] considered double pyramidal centralconfigurations for Newtonian 119873 + 2-body problems Melloand Fernandes [12] analyzed new classes of spatial centralconfigurations for the119873+ 3-body problem Llibre and Mellostudied triple and quadruple nested central configurations forthe planar 119899-body problem There are many papers studyingcentral configuration problems such as [13ndash22]
Based on the above works we study stacked centralconfiguration for Newtonian 119873 + 2119901 + 1-body problems Inthe119873+2119901+1-body problems119873 bodies are at the vertices ofa regular 119873-gon 119879 and 2119901 bodies are symmetrically locatedon the same straight line which is perpendicular to 119879 andpasses through the center of 119879 the 119873 + 2119901 + 1th body islocated at the center of 119879 The masses located on the verticesof the regular119873-gon are equal the masses located on the lineand symmetric with respect to the center of 119879 are equal (seeFigure 1 for119873 = 4 and 119901 = 2)
In this paper we will prove the following result
Theorem 2 For119873+ 2119901 + 1-body problem in 1198773 where119873 ge 2and 119901 ge 1 there is at least one central configuration such that119873 bodies are at the vertices of a regular119873-gon119879 and 2119901 bodiesare symmetric with respect to the center of the regular 119873-gon119879 and located on a line which is perpendicular to the regular119873-gon119879 the119873+2119901+1th body is located at the center of119879Themasses at the vertices of 119879 are equal and the masses symmetricwith respect to the center of 119879 are equal
2 The Proof of Theorem 2
Our approach to Theorem 2 is inspired by the of argumentsof Corbera et al in [23]
21 Equations for the Central Configurations of 119873 + 2119901-Body Problems To begin we take a coordinate system whichsimplifies the analysis The particles have positions given by119902119895 = (cos120572119895 sin120572119895 0) where 120572119895 = ((119895 minus 1)119873)2120587 119895 =1 119873 119902119873+119895 = (0 0 119903119895) 119902119873+119895+119901 = (0 0 minus119903119895) where 119895 =1 119901 119902119873+2119901+1 = (0 0 0)
The masses are given by 1198981 = 1198982 = sdot sdot sdot = 119898119873 = 1119898119873+119895 = 119898119873+119895+119901 = 119872119895 where 119895 = 1 119901 119898119873+2119901+1 = 1198720Notice that (1199021 119902119873 119902119873+1 119902119873+2119901 119902119873+2119901+1) is a cen-
tral configuration if and only if
119873+2119901+1sum119895=1119895 =119896
11989811989511989811989610038161003816100381610038161003816119902119895 minus 119902119896100381610038161003816100381610038163 (119902119895 minus 119902119896) = minus120582119898119896119902119896 1 ⩽ 119896 ⩽ 119873 + 2119901(7)
By the symmetries of the system (7) is equivalent to thefollowing equations
119873+2119901+1sum119895=1119895 =119896
11989811989511989811989610038161003816100381610038161003816119902119895 minus 119902119896100381610038161003816100381610038163 (119902119895 minus 119902119896) = minus120582119898119896119902119896119896 = 1119873 + 1 119873 + 119901 (8)
that is minus120582 (1 0 0) = minus 120573 (1 0 0) minus (1 0 0)1198720minus 119901sum119894=1
210038161003816100381610038161 + 1199032119894 100381610038161003816100381632119872119894 (1 0 0) (9)
where 120573 = (14)sum119873minus1119895=1 csc(120587119895119873)minus120582 (0 0 1199031) = minus 11987310038161003816100381610038161 + 11990321 100381610038161003816100381632 (0 0 1199031) minus (0 0 111990321 )1198720minus 1100381610038161003816100381621199031100381610038161003816100381621198721 (0 0 1)+ 119901sum119894=2
[ 11003816100381610038161003816119903119894 minus 119903110038161003816100381610038162 minus 11003816100381610038161003816119903119894 + 119903110038161003816100381610038162]1198721 (0 0 1) minus120582 (0 0 119903119895) = minus 119873100381610038161003816100381610038161 + 1199032119895 1003816100381610038161003816100381632 (0 0 119903119895) minus (0 0 11199032119895 )1198720minus sum1le119894lt119895
( 110038161003816100381610038161003816119903119894 minus 119903119895100381610038161003816100381610038162 + 110038161003816100381610038161003816119903119894 + 119903119895100381610038161003816100381610038162)119872119894 (0 0 1)minus 1100381610038161003816100381610038162119903119895100381610038161003816100381610038162119872119895 (0 0 1)+ sum119895lt119894le119901
( 110038161003816100381610038161003816119903119894 minus 119903119895100381610038161003816100381610038162 minus 110038161003816100381610038161003816119903119894 + 119903119895100381610038161003816100381610038162)119872119894 (0 0 1) 119895 = 2 119901 minus 1(10)
Abstract and Applied Analysis 3
minus120582 (0 0 119903119901) = minus 119873100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381632 (0 0 119903119901) minus (0 0 11199032119901)1198720minus sum1le119894lt119901
( 110038161003816100381610038161003816119903119894 minus 119903119901100381610038161003816100381610038162 + 110038161003816100381610038161003816119903119894 + 119903119901100381610038161003816100381610038162)119872119894 (0 0 1)minus 1100381610038161003816100381610038162119903119901100381610038161003816100381610038162119872119901 (0 0 1)
(11)
In order to simplify the equations we defined1198860119895 = minus2|1+1199032119895 |32 119886119895119895 = minus141199033119895 where 119895 = 1 119901119886119894119895 = (1|119903119894 minus 119903119895|2119903119894 minus 1|119903119894 + 119903119895|2119903119894) when 119894 lt 119895119886119894119895 = minus(1|119903119894 minus 119903119895|2119903119894 + 1|119903119894 + 119903119895|2119903119894) when 119894 gt 11989511988600 = 1198861198940 = 1 when 119894 = 1 1199011198870 = 120573 + 1198720 119887119894 = 119873|1 + 1199032119894 |32 + 11987201199033119894 where 119894 =1 119901Equations (9)ndash(11) can be written as a linear system of theform 119860119883 = 119887 given by
(1 11988601 11988602 11988603 sdot sdot sdot 11988601199011 11988611 11988612 11988613 sdot sdot sdot 11988611199011 11988621 11988622 11988623 sdot sdot sdot 1198862119901
1 1198861199011 1198861199012 1198861199013 sdot sdot sdot 119886119901119901)(((
120582119872111987221198723119872119875)))
=(((
1198870119887111988721198873119887119901)))
(12)
The column vector is given by the variables 119883 = (12058211987211198722 119872119901)119879 Since the 119886119894119895 is function of 1199031 1199032 119903119901 wewrite the coefficient matrix as 119860119901(1199031 1199032 119903119901)22 For 119901 = 1 We need the next lemma
Lemma 3 (see [12]) Assuming 1198981 = sdot sdot sdot = 119898119873 =1 119898119873+1 = 119898119873+2 = 1198721 there is a nonempty interval119868 sub 119877 1198720(1199031) and 1198721(1199031) such that for each 1199031 isin 119868(1199021 119902119873 119902119873+1 119902119873+2 119902119873+3) forms a central configuration ofthe119873 + 2 + 1-body problem
For 119901 = 1 system (12) becomes(1 119886011 11988611)( 1205821198721) = (11988701198871) (13)10038161003816100381610038161198601 (1199031)1003816100381610038161003816 = 11988611 minus 11988601 = minus 1411990331 + 210038161003816100381610038161 + 11990321 100381610038161003816100381632 (14)
If we consider |1198601(1199031)| as a function of 1199031 then |1198601(1199031)| is ananalytic function and nonconstant By Lemma 3 there exists
a r1 isin 119868 such that (13) has a unique solution (1205821198721) satisfying120582 gt 0 and1198721 gt 023 For All 119901gt1 The proof for 119901 ge 1 is done by inductionWe claim that there exists 0 lt 1199031 lt 1199032 lt sdot sdot sdot lt 119903119901 such thatsystem (12) has a unique solution 120582 = 120582(1199031 119903119901) gt 0119872119894 =119872119894(1199031 119903119901) gt 0 for 119894 = 1 119901 We have seen that theclaim is true for 119901 = 1 We assume the claim is true for 119901 minus 1and we will prove it for 119901 Assume by induction hypothesisthat there exists 0 lt 1 lt 2 lt sdot sdot sdot lt 119901minus1 such that system(12) has a unique solution = 120582(1 2 119901minus1) gt 0 and119894 = 119872119894(1 2 119901minus1) gt 0 for 119894 = 1 2 119901 minus 1
We need the next lemma
Lemma 4 There exists 119901 gt 119901minus1 such that = 120582(1 2 119901minus1) 119894 = 119872119894(1 2 119901minus1) for 119894 = 1 2 119901 minus 1 and119901 = 0 is a solution of (12)
Proof Since 119872119901 = 119901 = 0 we have that the first 119901 minus 1equation of (12) is satisfied when 120582 = 119872119894 = 119894 for 119894 =1 2 119901 minus 1 and119872119901 = 119901 = 0 Substituting this solutioninto the last equation of (12) we let119891 (119903119901) = + 11988611990111 + sdot sdot sdot + 119886119901119901minus1119901minus1 minus 119873100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381632 minus 11987201199033119901
(15)
We have that
lim119903119901rarr+infin
119891 (119903119901) = gt 0 lim119903119901rarr
+119901minus1
119891 (119903119901) = minusinfin (16)
Therefore there exists at least a value 119903119901 = 119901 gt 119901minus1 satisfyingequation 119891(119903119901) = 0 This completes the proof of Lemma 4
By using the implicit function theorem we will prove thatthe solution of (12) given in Lemma 3 can be continued to asolution with119872119901 gt 0
Let 119904 = (120582 1199031 1199031199011198721 119872119901) we define1198920 (119904) = 120582 + 119886011198721 + sdot sdot sdot + 1198860119901119872119901 minus 11988701198921 (119904) = 120582 + 119886111198721 + sdot sdot sdot + 1198861119901119872119901 minus 11988711198922 (119904) = 120582 + 119886211198721 + sdot sdot sdot + 1198862119901119872119901 minus 1198872119892119901 (119904) = 120582 + 11988611990111198721 + sdot sdot sdot + 119886119901119901119872119901 minus 119887119901
(17)
It is not difficult to see that the system (12) is equivalent to119892119894(119904) = 0 for 119894 = 0 1 119901Let = ( 1 2 119901 1 119901) be the solution of
system (12) given in Lemma 4 The differential of (17) with
4 Abstract and Applied Analysis
respect to the variables (12058211987211198722 119872119901minus1 119903119901) is119863119901 () =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 11988601 11988602 sdot sdot sdot 1198860119901minus1 01 11988611 11988612 sdot sdot sdot 1198861119901minus1 0
1 119886119901minus11 119886119901minus12 sdot sdot sdot 119886119901minus1119901minus1 01 1198861199011 1198861199012 sdot sdot sdot 119886119901119901minus1 120597119892119901 ()1205971199031199011003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816119860119901minus1 (1 2 119901minus1)10038161003816100381610038161003816 120597119892119901 ()120597119903119901 120597119892119901 ()120597119903119901 = 119901minus1sum
119894=1
[[ 210038161003816100381610038161003816119903119901 minus 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 minus 1199031198941003816100381610038161003816100381621199032119901+ 210038161003816100381610038161003816119903119901 + 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 + 1199031198941003816100381610038161003816100381631199032119901]]119894+ (3119873119903119901)100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381652 + 311987201199034119901 gt 0
(18)
Wehave assumed that 0 lt 1 lt 2 lt sdot sdot sdot lt 119901minus1exists such thatsystem (12) with 119901 minus 1 instead of 119901 has a unique solution so|119860119901minus1(1 2 119901minus1)| = 0 therefore 119863() = 0 Applying theImplicit Function Theorem there exists a neighborhood 119880of (119901 1 1 119901minus1) and unique analytic functions 120582 =120582(119901 1 1 119901minus1) 119872119894 = 119872119894(119901 1 1 119901minus1) for 119894 =1 119901 minus 1 and 119903119901 = 119903119901(119872119901 1199031 1199032 119903119901minus1) such that(12058211987211198722 119872119901) is the solution of the system (12) for all(119872119901 1199031 1199032 119903119901minus1) isin 119880 The determinant is calculated as10038161003816100381610038161003816119860119901 (1199031 1199032 119903119901)10038161003816100381610038161003816= 119886119901119901 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum
119894=0
119886119901119894119861119901119894= minus1100381610038161003816100381610038162119903119901100381610038161003816100381610038162 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum119894=0119886119901119894119861119901119894
(19)
where 119861119901119894 is the algebraic cofactor of 119886119901119894We see that |119860119901minus1(1199031 1199032 119903119901minus1)| 119886119901119894 and 119861119901119894 for119894 = 0 1 119901 minus 1 do not contain the factor 1|2119903119901|2
If we consider |119860119901(1199031 1199032 119903119901)| as a function of 119903119901then |119860119901(1199031 1199032 119903119901)| is analytic and nonconstant Wecan find (1199031 1199032 119903119901) sufficiently close to (1 2 119901)such that |119860119901(1199031 1199032 119903119901)| = 0 and therefore a solution(1205821198721 119872119901) of system (12) is satisfying 120582 gt 0 119872119894 gt 0for 119894 = 1 119901
The proof of Theorem 2 is completed
Acknowledgments
The authors express their gratitude to Professor ZhangShiqing for his discussions and helpful suggestionsThiswork
is supported by NSF of China and Youth found of MianyangNormal University
References
[1] R Abraham and J E Marsden Foundation of MechanicsBenjamin New York NY USA 2nd edition 1978
[2] D G Saari Rings and Other Newtonian N-body ProblemsAmerican Mathematical Society Providence RI USA 2005
[3] A Wintner The Analytical Foundations of Celestial Mechanicsvol 5 of Princeton Mathematical Series Princeton UniversityPress Princeton NJ USA 1941
[4] D G Saari ldquoSingularities and collisions of Newtonian gravi-tational systemsrdquo Archive for Rational Mechanics and Analysisvol 49 pp 311ndash320 1973
[5] D G Saari ldquoOn the role and the properties of n-body centralconfigurationsrdquo Celestial Mechanics and Dynamical Astronomyvol 21 no 1 pp 9ndash20 1980
[6] D G Saari and N D Hulkower ldquoOn the manifolds of totalcollapse orbits and of completely parabolic orbits for the n-bodyproblemrdquo Journal of Differential Equations vol 41 no 1 pp 27ndash43 1981
[7] S Smale ldquoMathematical problems for the next centuryrdquoMath-ematical Intelligencer vol 20 pp 141ndash145 1998
[8] M Hampton and R Moeckel ldquoFiniteness of relative equilibriaof the four-body problemrdquo Inventiones Mathematicae vol 163no 2 pp 289ndash312 2006
[9] M Hampton ldquoStacked central configurations new examples inthe planar five-body problemrdquo Nonlinearity vol 18 no 5 pp2299ndash2304 2005
[10] T Ouyang Z Xie and S Zhang ldquoPyramidal central configura-tions and perverse solutionsrdquo Electronic Journal of DifferentialEquations vol 2004 no 106 pp 1ndash9 2004
[11] S Zhang and Q Zhou ldquoDouble pyramidal central configura-tionsrdquo Physics Letters A vol 281 no 4 pp 240ndash248 2001
[12] L F Mello and A C Fernandes ldquoNew classes of spatial centralconfigurations for the n-body problemrdquo Nonlinear AnalysisReal World Applications vol 12 no 1 pp 723ndash730 2011
[13] A Albouy ldquoThe symmetric central configurations of four equalmassesrdquo in Hamiltonian Dynamics and Celestial Mechanicsvol 198 of Contemporary Mathematics pp 131ndash135 AmericanMathematical Society Providence RI USA 1996
[14] A Albouy Y Fu and S Sun ldquoSymmetry of planar four-bodyconvex central configurationsrdquo Proceedings of The Royal Societyof London Series A vol 464 no 2093 pp 1355ndash1365 2008
[15] F N Diacu ldquoThe masses in a symmetric centered solution ofthe n-body problemrdquo Proceedings of the AmericanMathematicalSociety vol 109 no 4 pp 1079ndash1085 1990
[16] J Lei and M Santoprete ldquoRosette central configurationsdegenerate central configurations and bifurcationsrdquo CelestialMechanics amp Dynamical Astronomy vol 94 no 3 pp 271ndash2872006
[17] R Moeckel ldquoOn central configurationsrdquo MathematischeZeitschrift vol 205 no 4 pp 499ndash517 1990
[18] R Moeckel and C Simo ldquoBifurcation of spatial central con-figurations from planar onesrdquo SIAM Journal on MathematicalAnalysis vol 26 no 4 pp 978ndash998 1995
[19] FR Moulton ldquoThe straight line solutions of the n-body prob-lemrdquoAnnals ofMathematics Second Series vol 12 pp 1ndash17 1910
Abstract and Applied Analysis 5
[20] L M Perko and E L Walter ldquoRegular polygon solutions of theN-body problemrdquo Proceedings of the American MathematicalSociety vol 94 no 2 pp 301ndash309 1985
[21] J Shi and Z Xie ldquoClassification of four-body central configura-tions with three equal massesrdquo Journal ofMathematical Analysisand Applications vol 363 no 2 pp 512ndash524 2010
[22] Z Xia ldquoCentral configurations with many small massesrdquoJournal of Differential Equations vol 91 no 1 pp 168ndash179 1991
[23] M Corbera J Delgado and J Llibre ldquoOn the existence ofcentral configurations of p nested n-gonsrdquo Qualitative Theoryof Dynamical Systems vol 8 no 2 pp 255ndash265 2009
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
minus120582 (0 0 119903119901) = minus 119873100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381632 (0 0 119903119901) minus (0 0 11199032119901)1198720minus sum1le119894lt119901
( 110038161003816100381610038161003816119903119894 minus 119903119901100381610038161003816100381610038162 + 110038161003816100381610038161003816119903119894 + 119903119901100381610038161003816100381610038162)119872119894 (0 0 1)minus 1100381610038161003816100381610038162119903119901100381610038161003816100381610038162119872119901 (0 0 1)
(11)
In order to simplify the equations we defined1198860119895 = minus2|1+1199032119895 |32 119886119895119895 = minus141199033119895 where 119895 = 1 119901119886119894119895 = (1|119903119894 minus 119903119895|2119903119894 minus 1|119903119894 + 119903119895|2119903119894) when 119894 lt 119895119886119894119895 = minus(1|119903119894 minus 119903119895|2119903119894 + 1|119903119894 + 119903119895|2119903119894) when 119894 gt 11989511988600 = 1198861198940 = 1 when 119894 = 1 1199011198870 = 120573 + 1198720 119887119894 = 119873|1 + 1199032119894 |32 + 11987201199033119894 where 119894 =1 119901Equations (9)ndash(11) can be written as a linear system of theform 119860119883 = 119887 given by
(1 11988601 11988602 11988603 sdot sdot sdot 11988601199011 11988611 11988612 11988613 sdot sdot sdot 11988611199011 11988621 11988622 11988623 sdot sdot sdot 1198862119901
1 1198861199011 1198861199012 1198861199013 sdot sdot sdot 119886119901119901)(((
120582119872111987221198723119872119875)))
=(((
1198870119887111988721198873119887119901)))
(12)
The column vector is given by the variables 119883 = (12058211987211198722 119872119901)119879 Since the 119886119894119895 is function of 1199031 1199032 119903119901 wewrite the coefficient matrix as 119860119901(1199031 1199032 119903119901)22 For 119901 = 1 We need the next lemma
Lemma 3 (see [12]) Assuming 1198981 = sdot sdot sdot = 119898119873 =1 119898119873+1 = 119898119873+2 = 1198721 there is a nonempty interval119868 sub 119877 1198720(1199031) and 1198721(1199031) such that for each 1199031 isin 119868(1199021 119902119873 119902119873+1 119902119873+2 119902119873+3) forms a central configuration ofthe119873 + 2 + 1-body problem
For 119901 = 1 system (12) becomes(1 119886011 11988611)( 1205821198721) = (11988701198871) (13)10038161003816100381610038161198601 (1199031)1003816100381610038161003816 = 11988611 minus 11988601 = minus 1411990331 + 210038161003816100381610038161 + 11990321 100381610038161003816100381632 (14)
If we consider |1198601(1199031)| as a function of 1199031 then |1198601(1199031)| is ananalytic function and nonconstant By Lemma 3 there exists
a r1 isin 119868 such that (13) has a unique solution (1205821198721) satisfying120582 gt 0 and1198721 gt 023 For All 119901gt1 The proof for 119901 ge 1 is done by inductionWe claim that there exists 0 lt 1199031 lt 1199032 lt sdot sdot sdot lt 119903119901 such thatsystem (12) has a unique solution 120582 = 120582(1199031 119903119901) gt 0119872119894 =119872119894(1199031 119903119901) gt 0 for 119894 = 1 119901 We have seen that theclaim is true for 119901 = 1 We assume the claim is true for 119901 minus 1and we will prove it for 119901 Assume by induction hypothesisthat there exists 0 lt 1 lt 2 lt sdot sdot sdot lt 119901minus1 such that system(12) has a unique solution = 120582(1 2 119901minus1) gt 0 and119894 = 119872119894(1 2 119901minus1) gt 0 for 119894 = 1 2 119901 minus 1
We need the next lemma
Lemma 4 There exists 119901 gt 119901minus1 such that = 120582(1 2 119901minus1) 119894 = 119872119894(1 2 119901minus1) for 119894 = 1 2 119901 minus 1 and119901 = 0 is a solution of (12)
Proof Since 119872119901 = 119901 = 0 we have that the first 119901 minus 1equation of (12) is satisfied when 120582 = 119872119894 = 119894 for 119894 =1 2 119901 minus 1 and119872119901 = 119901 = 0 Substituting this solutioninto the last equation of (12) we let119891 (119903119901) = + 11988611990111 + sdot sdot sdot + 119886119901119901minus1119901minus1 minus 119873100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381632 minus 11987201199033119901
(15)
We have that
lim119903119901rarr+infin
119891 (119903119901) = gt 0 lim119903119901rarr
+119901minus1
119891 (119903119901) = minusinfin (16)
Therefore there exists at least a value 119903119901 = 119901 gt 119901minus1 satisfyingequation 119891(119903119901) = 0 This completes the proof of Lemma 4
By using the implicit function theorem we will prove thatthe solution of (12) given in Lemma 3 can be continued to asolution with119872119901 gt 0
Let 119904 = (120582 1199031 1199031199011198721 119872119901) we define1198920 (119904) = 120582 + 119886011198721 + sdot sdot sdot + 1198860119901119872119901 minus 11988701198921 (119904) = 120582 + 119886111198721 + sdot sdot sdot + 1198861119901119872119901 minus 11988711198922 (119904) = 120582 + 119886211198721 + sdot sdot sdot + 1198862119901119872119901 minus 1198872119892119901 (119904) = 120582 + 11988611990111198721 + sdot sdot sdot + 119886119901119901119872119901 minus 119887119901
(17)
It is not difficult to see that the system (12) is equivalent to119892119894(119904) = 0 for 119894 = 0 1 119901Let = ( 1 2 119901 1 119901) be the solution of
system (12) given in Lemma 4 The differential of (17) with
4 Abstract and Applied Analysis
respect to the variables (12058211987211198722 119872119901minus1 119903119901) is119863119901 () =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 11988601 11988602 sdot sdot sdot 1198860119901minus1 01 11988611 11988612 sdot sdot sdot 1198861119901minus1 0
1 119886119901minus11 119886119901minus12 sdot sdot sdot 119886119901minus1119901minus1 01 1198861199011 1198861199012 sdot sdot sdot 119886119901119901minus1 120597119892119901 ()1205971199031199011003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816119860119901minus1 (1 2 119901minus1)10038161003816100381610038161003816 120597119892119901 ()120597119903119901 120597119892119901 ()120597119903119901 = 119901minus1sum
119894=1
[[ 210038161003816100381610038161003816119903119901 minus 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 minus 1199031198941003816100381610038161003816100381621199032119901+ 210038161003816100381610038161003816119903119901 + 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 + 1199031198941003816100381610038161003816100381631199032119901]]119894+ (3119873119903119901)100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381652 + 311987201199034119901 gt 0
(18)
Wehave assumed that 0 lt 1 lt 2 lt sdot sdot sdot lt 119901minus1exists such thatsystem (12) with 119901 minus 1 instead of 119901 has a unique solution so|119860119901minus1(1 2 119901minus1)| = 0 therefore 119863() = 0 Applying theImplicit Function Theorem there exists a neighborhood 119880of (119901 1 1 119901minus1) and unique analytic functions 120582 =120582(119901 1 1 119901minus1) 119872119894 = 119872119894(119901 1 1 119901minus1) for 119894 =1 119901 minus 1 and 119903119901 = 119903119901(119872119901 1199031 1199032 119903119901minus1) such that(12058211987211198722 119872119901) is the solution of the system (12) for all(119872119901 1199031 1199032 119903119901minus1) isin 119880 The determinant is calculated as10038161003816100381610038161003816119860119901 (1199031 1199032 119903119901)10038161003816100381610038161003816= 119886119901119901 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum
119894=0
119886119901119894119861119901119894= minus1100381610038161003816100381610038162119903119901100381610038161003816100381610038162 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum119894=0119886119901119894119861119901119894
(19)
where 119861119901119894 is the algebraic cofactor of 119886119901119894We see that |119860119901minus1(1199031 1199032 119903119901minus1)| 119886119901119894 and 119861119901119894 for119894 = 0 1 119901 minus 1 do not contain the factor 1|2119903119901|2
If we consider |119860119901(1199031 1199032 119903119901)| as a function of 119903119901then |119860119901(1199031 1199032 119903119901)| is analytic and nonconstant Wecan find (1199031 1199032 119903119901) sufficiently close to (1 2 119901)such that |119860119901(1199031 1199032 119903119901)| = 0 and therefore a solution(1205821198721 119872119901) of system (12) is satisfying 120582 gt 0 119872119894 gt 0for 119894 = 1 119901
The proof of Theorem 2 is completed
Acknowledgments
The authors express their gratitude to Professor ZhangShiqing for his discussions and helpful suggestionsThiswork
is supported by NSF of China and Youth found of MianyangNormal University
References
[1] R Abraham and J E Marsden Foundation of MechanicsBenjamin New York NY USA 2nd edition 1978
[2] D G Saari Rings and Other Newtonian N-body ProblemsAmerican Mathematical Society Providence RI USA 2005
[3] A Wintner The Analytical Foundations of Celestial Mechanicsvol 5 of Princeton Mathematical Series Princeton UniversityPress Princeton NJ USA 1941
[4] D G Saari ldquoSingularities and collisions of Newtonian gravi-tational systemsrdquo Archive for Rational Mechanics and Analysisvol 49 pp 311ndash320 1973
[5] D G Saari ldquoOn the role and the properties of n-body centralconfigurationsrdquo Celestial Mechanics and Dynamical Astronomyvol 21 no 1 pp 9ndash20 1980
[6] D G Saari and N D Hulkower ldquoOn the manifolds of totalcollapse orbits and of completely parabolic orbits for the n-bodyproblemrdquo Journal of Differential Equations vol 41 no 1 pp 27ndash43 1981
[7] S Smale ldquoMathematical problems for the next centuryrdquoMath-ematical Intelligencer vol 20 pp 141ndash145 1998
[8] M Hampton and R Moeckel ldquoFiniteness of relative equilibriaof the four-body problemrdquo Inventiones Mathematicae vol 163no 2 pp 289ndash312 2006
[9] M Hampton ldquoStacked central configurations new examples inthe planar five-body problemrdquo Nonlinearity vol 18 no 5 pp2299ndash2304 2005
[10] T Ouyang Z Xie and S Zhang ldquoPyramidal central configura-tions and perverse solutionsrdquo Electronic Journal of DifferentialEquations vol 2004 no 106 pp 1ndash9 2004
[11] S Zhang and Q Zhou ldquoDouble pyramidal central configura-tionsrdquo Physics Letters A vol 281 no 4 pp 240ndash248 2001
[12] L F Mello and A C Fernandes ldquoNew classes of spatial centralconfigurations for the n-body problemrdquo Nonlinear AnalysisReal World Applications vol 12 no 1 pp 723ndash730 2011
[13] A Albouy ldquoThe symmetric central configurations of four equalmassesrdquo in Hamiltonian Dynamics and Celestial Mechanicsvol 198 of Contemporary Mathematics pp 131ndash135 AmericanMathematical Society Providence RI USA 1996
[14] A Albouy Y Fu and S Sun ldquoSymmetry of planar four-bodyconvex central configurationsrdquo Proceedings of The Royal Societyof London Series A vol 464 no 2093 pp 1355ndash1365 2008
[15] F N Diacu ldquoThe masses in a symmetric centered solution ofthe n-body problemrdquo Proceedings of the AmericanMathematicalSociety vol 109 no 4 pp 1079ndash1085 1990
[16] J Lei and M Santoprete ldquoRosette central configurationsdegenerate central configurations and bifurcationsrdquo CelestialMechanics amp Dynamical Astronomy vol 94 no 3 pp 271ndash2872006
[17] R Moeckel ldquoOn central configurationsrdquo MathematischeZeitschrift vol 205 no 4 pp 499ndash517 1990
[18] R Moeckel and C Simo ldquoBifurcation of spatial central con-figurations from planar onesrdquo SIAM Journal on MathematicalAnalysis vol 26 no 4 pp 978ndash998 1995
[19] FR Moulton ldquoThe straight line solutions of the n-body prob-lemrdquoAnnals ofMathematics Second Series vol 12 pp 1ndash17 1910
Abstract and Applied Analysis 5
[20] L M Perko and E L Walter ldquoRegular polygon solutions of theN-body problemrdquo Proceedings of the American MathematicalSociety vol 94 no 2 pp 301ndash309 1985
[21] J Shi and Z Xie ldquoClassification of four-body central configura-tions with three equal massesrdquo Journal ofMathematical Analysisand Applications vol 363 no 2 pp 512ndash524 2010
[22] Z Xia ldquoCentral configurations with many small massesrdquoJournal of Differential Equations vol 91 no 1 pp 168ndash179 1991
[23] M Corbera J Delgado and J Llibre ldquoOn the existence ofcentral configurations of p nested n-gonsrdquo Qualitative Theoryof Dynamical Systems vol 8 no 2 pp 255ndash265 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
respect to the variables (12058211987211198722 119872119901minus1 119903119901) is119863119901 () =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 11988601 11988602 sdot sdot sdot 1198860119901minus1 01 11988611 11988612 sdot sdot sdot 1198861119901minus1 0
1 119886119901minus11 119886119901minus12 sdot sdot sdot 119886119901minus1119901minus1 01 1198861199011 1198861199012 sdot sdot sdot 119886119901119901minus1 120597119892119901 ()1205971199031199011003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816119860119901minus1 (1 2 119901minus1)10038161003816100381610038161003816 120597119892119901 ()120597119903119901 120597119892119901 ()120597119903119901 = 119901minus1sum
119894=1
[[ 210038161003816100381610038161003816119903119901 minus 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 minus 1199031198941003816100381610038161003816100381621199032119901+ 210038161003816100381610038161003816119903119901 + 119903119894100381610038161003816100381610038163119903119901 + 110038161003816100381610038161003816119903119901 + 1199031198941003816100381610038161003816100381631199032119901]]119894+ (3119873119903119901)100381610038161003816100381610038161 + 11990321199011003816100381610038161003816100381652 + 311987201199034119901 gt 0
(18)
Wehave assumed that 0 lt 1 lt 2 lt sdot sdot sdot lt 119901minus1exists such thatsystem (12) with 119901 minus 1 instead of 119901 has a unique solution so|119860119901minus1(1 2 119901minus1)| = 0 therefore 119863() = 0 Applying theImplicit Function Theorem there exists a neighborhood 119880of (119901 1 1 119901minus1) and unique analytic functions 120582 =120582(119901 1 1 119901minus1) 119872119894 = 119872119894(119901 1 1 119901minus1) for 119894 =1 119901 minus 1 and 119903119901 = 119903119901(119872119901 1199031 1199032 119903119901minus1) such that(12058211987211198722 119872119901) is the solution of the system (12) for all(119872119901 1199031 1199032 119903119901minus1) isin 119880 The determinant is calculated as10038161003816100381610038161003816119860119901 (1199031 1199032 119903119901)10038161003816100381610038161003816= 119886119901119901 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum
119894=0
119886119901119894119861119901119894= minus1100381610038161003816100381610038162119903119901100381610038161003816100381610038162 10038161003816100381610038161003816119860119901minus1 (1199031 1199032 119903119901minus1)10038161003816100381610038161003816 + 119901minus1sum119894=0119886119901119894119861119901119894
(19)
where 119861119901119894 is the algebraic cofactor of 119886119901119894We see that |119860119901minus1(1199031 1199032 119903119901minus1)| 119886119901119894 and 119861119901119894 for119894 = 0 1 119901 minus 1 do not contain the factor 1|2119903119901|2
If we consider |119860119901(1199031 1199032 119903119901)| as a function of 119903119901then |119860119901(1199031 1199032 119903119901)| is analytic and nonconstant Wecan find (1199031 1199032 119903119901) sufficiently close to (1 2 119901)such that |119860119901(1199031 1199032 119903119901)| = 0 and therefore a solution(1205821198721 119872119901) of system (12) is satisfying 120582 gt 0 119872119894 gt 0for 119894 = 1 119901
The proof of Theorem 2 is completed
Acknowledgments
The authors express their gratitude to Professor ZhangShiqing for his discussions and helpful suggestionsThiswork
is supported by NSF of China and Youth found of MianyangNormal University
References
[1] R Abraham and J E Marsden Foundation of MechanicsBenjamin New York NY USA 2nd edition 1978
[2] D G Saari Rings and Other Newtonian N-body ProblemsAmerican Mathematical Society Providence RI USA 2005
[3] A Wintner The Analytical Foundations of Celestial Mechanicsvol 5 of Princeton Mathematical Series Princeton UniversityPress Princeton NJ USA 1941
[4] D G Saari ldquoSingularities and collisions of Newtonian gravi-tational systemsrdquo Archive for Rational Mechanics and Analysisvol 49 pp 311ndash320 1973
[5] D G Saari ldquoOn the role and the properties of n-body centralconfigurationsrdquo Celestial Mechanics and Dynamical Astronomyvol 21 no 1 pp 9ndash20 1980
[6] D G Saari and N D Hulkower ldquoOn the manifolds of totalcollapse orbits and of completely parabolic orbits for the n-bodyproblemrdquo Journal of Differential Equations vol 41 no 1 pp 27ndash43 1981
[7] S Smale ldquoMathematical problems for the next centuryrdquoMath-ematical Intelligencer vol 20 pp 141ndash145 1998
[8] M Hampton and R Moeckel ldquoFiniteness of relative equilibriaof the four-body problemrdquo Inventiones Mathematicae vol 163no 2 pp 289ndash312 2006
[9] M Hampton ldquoStacked central configurations new examples inthe planar five-body problemrdquo Nonlinearity vol 18 no 5 pp2299ndash2304 2005
[10] T Ouyang Z Xie and S Zhang ldquoPyramidal central configura-tions and perverse solutionsrdquo Electronic Journal of DifferentialEquations vol 2004 no 106 pp 1ndash9 2004
[11] S Zhang and Q Zhou ldquoDouble pyramidal central configura-tionsrdquo Physics Letters A vol 281 no 4 pp 240ndash248 2001
[12] L F Mello and A C Fernandes ldquoNew classes of spatial centralconfigurations for the n-body problemrdquo Nonlinear AnalysisReal World Applications vol 12 no 1 pp 723ndash730 2011
[13] A Albouy ldquoThe symmetric central configurations of four equalmassesrdquo in Hamiltonian Dynamics and Celestial Mechanicsvol 198 of Contemporary Mathematics pp 131ndash135 AmericanMathematical Society Providence RI USA 1996
[14] A Albouy Y Fu and S Sun ldquoSymmetry of planar four-bodyconvex central configurationsrdquo Proceedings of The Royal Societyof London Series A vol 464 no 2093 pp 1355ndash1365 2008
[15] F N Diacu ldquoThe masses in a symmetric centered solution ofthe n-body problemrdquo Proceedings of the AmericanMathematicalSociety vol 109 no 4 pp 1079ndash1085 1990
[16] J Lei and M Santoprete ldquoRosette central configurationsdegenerate central configurations and bifurcationsrdquo CelestialMechanics amp Dynamical Astronomy vol 94 no 3 pp 271ndash2872006
[17] R Moeckel ldquoOn central configurationsrdquo MathematischeZeitschrift vol 205 no 4 pp 499ndash517 1990
[18] R Moeckel and C Simo ldquoBifurcation of spatial central con-figurations from planar onesrdquo SIAM Journal on MathematicalAnalysis vol 26 no 4 pp 978ndash998 1995
[19] FR Moulton ldquoThe straight line solutions of the n-body prob-lemrdquoAnnals ofMathematics Second Series vol 12 pp 1ndash17 1910
Abstract and Applied Analysis 5
[20] L M Perko and E L Walter ldquoRegular polygon solutions of theN-body problemrdquo Proceedings of the American MathematicalSociety vol 94 no 2 pp 301ndash309 1985
[21] J Shi and Z Xie ldquoClassification of four-body central configura-tions with three equal massesrdquo Journal ofMathematical Analysisand Applications vol 363 no 2 pp 512ndash524 2010
[22] Z Xia ldquoCentral configurations with many small massesrdquoJournal of Differential Equations vol 91 no 1 pp 168ndash179 1991
[23] M Corbera J Delgado and J Llibre ldquoOn the existence ofcentral configurations of p nested n-gonsrdquo Qualitative Theoryof Dynamical Systems vol 8 no 2 pp 255ndash265 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
[20] L M Perko and E L Walter ldquoRegular polygon solutions of theN-body problemrdquo Proceedings of the American MathematicalSociety vol 94 no 2 pp 301ndash309 1985
[21] J Shi and Z Xie ldquoClassification of four-body central configura-tions with three equal massesrdquo Journal ofMathematical Analysisand Applications vol 363 no 2 pp 512ndash524 2010
[22] Z Xia ldquoCentral configurations with many small massesrdquoJournal of Differential Equations vol 91 no 1 pp 168ndash179 1991
[23] M Corbera J Delgado and J Llibre ldquoOn the existence ofcentral configurations of p nested n-gonsrdquo Qualitative Theoryof Dynamical Systems vol 8 no 2 pp 255ndash265 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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