We consider a system of five mass points r1, r2, r3, and r4 with masses m1 = m2 = m and

$m_3= m_4= \tilde{m}$
m3=m4=m̃ moving about a single massive body r0 with mass m0 at its center which is assumed to be the origin of the coordinates system. We assume that the central body r0 makes a generalized force on the four mass points and that such a force is generated by a Manev's type potential, i.e., characterized by a potential of the form
$\frac{1}{r}+\frac{\epsilon }{r^2}$
1r+εr2⁠, on the other hand is assumed that the attraction between the bodies r1, r2, r3, and r4 is of the Newtonian type. This model represents several cases, for instance, when the central body is a spheroid or a radiating source. First, we prove the existence of three different relative rhomboidal solutions, and its (central) configuration is as follows: (1) the rhombus is a square and all primaries have equal masses; (2) the rhombus is not a square but all masses are equal; and (3) the rhombus is not a square and the pairs of primaries have different masses. The first two cases present two parameters: ε, the radiation or the oblateness coefficient and m = μ the common mass of the primaries. In the third case, the only parameter to be considered is ε since it can be shown that in this case both m and
$\tilde{m}$
m̃ depend on ε. Second, we study the stability of these solutions. We determine the values of the parameters (ε, μ) for which the square solution is spectrally stable and we prove that the rhomboidal solutions are unstable in the Lyapunov sense in the other two situations.

Topics
Linear stability analysis, Relativistic corrections, Hamiltonian mechanics, Central-force, Stellar structure and properties, Celestial mechanics, Coordinate system, Recurrence relations, Matrix calculus, Complex analysis
1.
Albouy
,
A.
, “
The symmetric central configurations of four equal masses
,”
Contemp. Math.
 
198
,
131
–
135
(
1995
).
https://doi.org/10.1090/conm/198
Google Scholar
Crossref
Search ADS
 
2.
Albouy
,
A.
,
Fu
,
Y.
, and
Sun
,
S.
, “
Symmetry of planar four-body convex central configurations
,”
Proc. R. Soc. London, Ser. A
 
464
(
2093
),
1355
–
1365
(
2008
).
https://doi.org/10.1098/rspa.2007.0320
Google Scholar
Crossref
Search ADS
 
3.
Arribas
,
M.
 and
Elipe
,
A.
, “
Bifurcations and equilibria in the extended N-body ring problem
,”
Mech. Res. Commun.
 
31
(
1
),
1
–
18
(
2004
).
https://doi.org/10.1016/S0093-6413(03)00086-7
Google Scholar
Crossref
Search ADS
 
4.
Arribas
,
M.
,
Elipe
,
A.
, and
Palacios
,
M.
, “
Linear stability of ring systems with generalized central forces
,”
Astron. Astrophys.
 
489
,
819
–
824
(
2008
).
https://doi.org/10.1051/0004-6361:200809903
Google Scholar
Crossref
Search ADS
 
5.
Arribas
,
M.
,
Elipe
,
A.
,
Kalvouridis
,
T.
, and
Palacios
,
M.
, “
Homographic solutions in the planar n + 1 body problem with quasi-homogeneous potentials
,”
Celest. Mech. Dyn. Astron.
 
99
(
1
),
1
–
12
(
2007
).
https://doi.org/10.1007/s10569-007-9083-8
Google Scholar
Crossref
Search ADS
 
6.
Corbera
,
M.
 and
Llibre
,
J.
, “
Infinitely many periodic orbits for the rhomboidal five-body problem
,”
J. Math. Phys.
 
47
,
122701
(
2006
).
https://doi.org/10.1063/1.2378617
Google Scholar
Crossref
Search ADS
 
7.
Craig
,
S.
,
Diacu
,
F.
,
Lacomba
,
E.
, and
Perez
,
E.
, “
On the anisotropic Manev problem
,”
J. Math. Phys.
 
40
,
1359
(
1999
).
https://doi.org/10.1063/1.532807
Google Scholar
Crossref
Search ADS
 
8.
Delgado
,
J.
,
Diacu
,
F.
,
Lacomba
,
E.
,
Mingarelli
,
A.
,
Mioc
,
V.
,
Perez
,
E.
, and
Stoica
,
C.
, “
The global flow of the Manev problem
,”
J. Math. Phys.
 
37
,
2748
(
1996
).
https://doi.org/10.1063/1.531539
Google Scholar
Crossref
Search ADS
 
9.
Diacu
,
F.
, “
Near-collision dynamics for particle systems with quasihomogeneous potentials
,”
J. Differ. Equations
 
128
,
58
–
77
(
1996
).
https://doi.org/10.1006/jdeq.1996.0089
Google Scholar
Crossref
Search ADS
 
10.
Diacu
,
F.
 and
Santoprete
,
M.
, “
On the global dynamics of the anisotropic Manev problem
,”
Physica D
 
194
,
75
–
94
(
2004
).
https://doi.org/10.1016/j.physd.2004.01.039
Google Scholar
Crossref
Search ADS
 
11.
Elipe
,
A.
,
Arribas
,
M.
, and
Kalvouridis
,
T.
, “
Central configuration in the planar n + 1 body problem with generalized forces
,” in
Proceedings of the Eighth Conference on Celestial Mechanics (Spanish)
[
Monogr. Real Acad. Cienc. Exactas, Fis., Quim. Nat. Zaragoza
 
28
,
1
–
8
(
2006
)].
Google Scholar
 
12.
Kulesza
,
M.
,
Marchesin
,
M.
, and
Vidal
,
C.
, “
Restricted rhomboidal five-body problem
,”
J. Phys. A
 
44
,
485204
(
2011
).
https://doi.org/10.1088/1751-8113/44/48/485204
Google Scholar
Crossref
Search ADS
 
13.
Maxwell
,
J. C.
,
On the Stability of Motions of Saturn's Rings
(
MacMillan
,
Cambridge
,
1859
).
Google Scholar
 
14.
Moeckel
,
R.
, “
Linear stability of relative equilibria with a domain mass
,”
J. Dyn. Differ. Equ.
 
6
(
1
),
37
–
51
(
1994
).
https://doi.org/10.1007/BF02219187
Google Scholar
Crossref
Search ADS
 
15.
Pendse
,
C. G.
,
Philos. Trans. R. Soc. London, Ser. A
 
CCXXXIV
,
145
(
1935
).
Google Scholar
 
16.
Roberts
,
G.
,
Monograph Series in Mathematics
(
World Scientific
,
2000
), Vol.
6
, p.
303
.
Google Scholar
 
17.
Santoprete
,
M.
, “
Symmetric periodic solutions of the anisotropic Manev problem
,”
J. Math. Phys.
 
43
,
3207
(
2002
).
https://doi.org/10.1063/1.1469670
Google Scholar
Crossref
Search ADS
 
18.
Scheeres
,
D. J.
 and
Vinh
,
N. X.
,
Celest. Mech. Dyn. Astron.
 
51
,
83
(
1991
).
https://doi.org/10.1007/BF02426672
Google Scholar
Crossref
Search ADS
 
19.
Sturm
,
J.
, “
Mémoire sur la résolution des équations numériques
,”
Bull. Sci. Frussac
 
11
,
419
–
425
(
1829
).
Google Scholar
 
20.
Szebehely
,
V.
,
Theory of Orbits
(
Academic Press
,
New York
,
1967
).
Google Scholar
 
21.
Tisserand
,
F.
,
Traité de Méchanique Céleste
(
Gauthier-Villars
,
Paris
,
1889
).
Google Scholar
 
22.
Vanberlei
,
R. J.
, “
Linear stability of ring systems around oblate central masses
,”
Adv. Space Res.
 
42
(
8
),
1370
–
1377
(
2008
).
https://doi.org/10.1016/j.asr.2008.01.001
Google Scholar
Crossref
Search ADS
 
23.
Vanberlei
,
R. J.
 and
Kolemen
,
E.
, “
Linear stability of ring systems
,”
Astron. J.
 
133
(
2
),
656
–
664
(
2007
).
https://doi.org/10.1086/510457
Google Scholar
Crossref
Search ADS
 
24.
Willerding
,
E.
, “
Theory of density waves in narrow planetary ring
,”
Astron. Astrophys.
 
161
(
2
),
403
–
407
(
1986
).
Google Scholar
 
25.
Wintner
,
A.
,
The Analytical Foundations of Celestial Mechanics
(
Princeton University Press
,
Princeton, NJ
,
1941
).
Google Scholar
 
© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.