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Astrophys Space Sci (2015) 358:3DOI 10.1007/s10509-015-2333-4
O R I G I NA L A RT I C L E
Nonlinear stability analysis in a equilateral restricted four-bodyproblem
Martha Alvarez-Ramírez1 · J.E.F. Skea2 · T.J. Stuchi3
Received: 5 December 2014 / Accepted: 26 March 2015 / Published online: 3 June 2015© Springer Science+Business Media Dordrecht 2015
Abstract We consider the problem of the stability of equi-librium points in the restricted equilateral four-body prob-lem where a particle of negligible mass is moving under theNewtonian gravitational attraction of three positive masses(called primaries) which move on circular orbits aroundtheir center of masses, such that their configuration is al-ways an equilateral triangle (Lagrangian configuration). Weconsider the case of two bodies of equal mass, which in adi-mensional units is the parameter of the problem. We inves-tigate the Birkhoff normal form around the linearly stableequilibrium points L3 for μ ∈ (0,0.0027096302) and, L5
and L6 for μ ∈ (0,0.018858526). The quadratic part of thenormalized Hamiltonian has no definite sign, so we examinea function which depends on the coefficients of the fourth or-der normal form and the stability follows from KAM theoryif this function is not zero, otherwise the normal form has tobe taken to sixth order or higher. This is the case for L5 andL6 for one value of μ. We calculate the sixth order normalform and establish the stability of the points in this case ofthe failure of the fourth order analysis. We also discuss thestability at the 2 : 1 resonances for the equilibrium point L3
and the symmetrical L5 and L6 points.
B J.E.F. [email protected]
M. Alvarez-Ramí[email protected]
T.J. [email protected]
1 Departamento de Matemáticas, UAM–Iztapalapa,09340 Iztapalapa, México, D.F., Mexico
2 Departamento de Física Teórica,Universidade do Estado do Rio de Janeiro,20550-900 Rio de Janeiro, RJ, Brazil
3 Universidade Federal do Rio de Janeiro,21941-909 Rio de Janeiro, RJ, Brazil
Keywords Four-body problem · Equilibrium points ·Stability · Birkhoff normal form
1 Introduction
The equilibrium points and their stability are an importantproperty of dynamical systems. It is well known that the sta-bility problem of equilibrium points in the n-body problemis a most difficult and intricate one. Lagrange, in 1772, stud-ied a special case of a three-body problem where the threebodies are placed at the corners of an equilateral triangle.He found a solution where the three bodies remain at con-stant distances from each other while they revolve aroundtheir common center of mass. Now, we know that the Trojanasteroids move near the triangular points of the Sun-Jupitersystem.
In order to derive stability properties for the four-bodyproblem, it is necessary to consider suitably simplified four-body systems. This paper is devoted to a planar equilateralrestricted four-body problem (in short, ERFBP), formulatedon the basis of Lagrange’s triangular solutions with twoequal masses. It can be cast as a Hamiltonian system of twodegrees of freedom, where the existence and the number ofcollinear and non-collinear equilibrium points depend on themass parameter of the primaries μ ∈ (0,1/2).
One example of the ERFBP is the Saturn-Tethys-Telesto(or Calypso)-spacecraft system where Tethys is the largestSaturn’s moon, and two small co-orbital satellites, Telestoand Calypso, lie at the L4 and L5 Lagrange points—60◦along the orbit, in front of and behind Tethys. See Fig. 1.
A number of studies exist on the restricted four-bodyproblem in this configuration: Alvarez-Ramírez and Vidal(2009) studied the ERFBP when the three primaries haveequal masses; Baltagiannis and Papadakis (2011) studied
3 Page 2 of 11 Astrophys Space Sci (2015) 358:3
Fig. 1 Saturn-Tethys-Telesto and Saturn-Tethys-Calypso configura-tion. This figure was taken from http://staff.on.br/jlkm/astron2e/AT_MEDIA/CH12/CHAP12AT.HTM
numerically the linear stability of the relative equilibriumsolutions for three cases depending on the masses of theprimaries, namely three equal masses, two primary bodieswith equal masses and all the primary bodies with unequalmasses. Besides, they showed the regions of the basins ofattraction for the equilibrium points for some values of themass parameters. Moreover, Ceccaroni and Biggs (2012) in-vestigated the stability of a ERFBP with three primaries inthe stable Lagrangian equilateral configuration and testedthe results in a real Sun-Jupiter-(624) Hektor-spacecraftsystem. They model incorporates “near term” low-thrustpropulsion capabilities to generate surfaces of artificial equi-librium points close to the smaller primary.
Budzko and Prokopenya (2011) have also studied thenonlinear stability of an particular equilibria in the sameERFBP; however, they considered that all the primary bod-ies have unequal masses, and solved the stability problem ina strict nonlinear formulation on the basis Arnold–Mosertheorem (see Markeev 1971). Peculiar properties of theHamiltonian normalization are discussed, as well as the in-fluence of the third and fourth order resonances on stabilityof the equilibrium positions.
More recently, Alvarez-Ramírez and Barrabés (2015)considered the case of two primaries of equal masses andstudied numerically the existence of families of unstable pe-riodic orbits, whose invariant stable and unstable manifoldsare responsible for the existence of homoclinic and hetero-clinic connections, as well as, of transit orbits traveling fromand to different regions.
From Alvarez-Ramírez and Barrabés (2015) and Bal-tagiannis and Papadakis (2011) we know that the exis-tence and the number of equilibrium points of the ERFBPwith two equal masses depend on the mass parameter μ,and that the number of collinear and non-collinear equi-librium points suffers bifurcation at μ∗ = 0.28827619129
and μ∗∗ = 0.44020149999, respectively. However, a com-plementary approach to the localization and stability of theequilibrium points is given here.
This paper investigates the nonlinear stability of collinearL3 and non-collinear L5,6 equilibrium points in the respec-tive range of the parameter, where they are elliptical equilib-rium points. In order to bring the Hamiltonian in a form suit-able for application of the normal form scheme, we performa sequence of transformations. We will move the origin ofthe coordinate system to the equilibrium point. We expandthe Hamiltonian function in Taylor series around the equi-librium point using the computer program Maple, whichallows us to compute the coefficients with accuracy up to32 decimal digits or more. Then we calculate the eigenval-ues and eigenvectors with which we construct a symplectictransformation that diagonalizes the quadratic part of theHamiltonian. Finally, following the Lie procedure, we ob-tain a Birkhoff normal form up to the fourth order, checkingwhether the quadratic part is of indefinite sign, we get anaccurate description of the dynamics in a (small enough)neighborhood of the equilibrium point, and then we applythe Arnold–Moser theorem, which establishes the stabilityof the equilibrium. All symbolic and numerical calculationsare done with an automatic procedure programmed.
This paper is organized as follows. In Sect. 2, we first in-troduce the planar restricted equilateral four-body problem,and the equations of motion are written in a synodical refer-ence frame. Then, in Sect. 3 we discuss the linear stabilityof the equilibrium solutions. In Sect. 4, we normalize theHamiltonian around an elliptic equilibrium points L3 andL5,6 to fourth order term, and we apply the Arnold–Mosertheorem to determine the stability of the equilibrium points.In Sect. 5 we obtain a third order resonant normal form tostudy the stability of 2 : 1 resonance for L3 and the symmet-rical points L5 and L6. In Sect. 6 we obtain a sixth order nor-mal form and determine the stability the equilibrium pointfor the value of μ where the forth order normal form fails.Finally, in the last section, some of the results are discussed.
2 Hamiltonian formulation
Consider three particles of masses m1, m2, m3 moving intheir mutual gravitational field. It is known that a particularsolution of the problem is an equilateral triangular configu-ration. The particles remain in the vertices of an equilateraltriangle and describe circular orbits around their center ofmass, which, by neglecting any external force, may be con-sidered at rest. Considering m1 + m2 + m3 = 1, the masseswill be m1 = 1 − 2μ and m2 = m3 = μ, where μ ∈ (0,1/2)
is called the mass parameter of the problem. See Fig. 2.We consider the motion of a infinitesimal particle m4, in
the gravitational field of m1, m2 and m3. As the primaries
Astrophys Space Sci (2015) 358:3 Page 3 of 11 3
Fig. 2 The equilateral restricted four-body problem
and m4 move in a plane, and the mass m4 is considered in-finitesimally small, the triangle configuration is not changed.
We take a rotating coordinate system (x, y) that rotateswith the uniform angular velocity of the primaries, so thatthey are fixed in the (x, y) plane. Then the coordinates ofthe primary bodies are
(x1, y1) = (√
3μ,0),
(x2, y2) =(
−√
3
2(1 − 2μ),
1
2
), (1)
(x3, y3) =(
−√
3
2(1 − 2μ),−1
2
).
The equations of motion of the infinitesimal mass in therotating coordinate system are (see, for example Baltagian-nis and Papadakis 2011)
x − 2y = Ωx,
y + 2x = Ωy,(2)
where
Ω = Ω(x,y) = 1
2
(x2 + y2) + 1 − 2μ
r1+ μ
r2+ μ
r3,
with r1 = √(x − x1)2 + y2, r2 =
√(x − x2)2 + (y − 1
2 )2
and r3 =√
(x − x2)2 + (y + 12 )2.
If we define the conjugate momenta as px = x − y andpy = y + x, then Hamiltonian is given as
H(x,y,px,py) = 1
2
(p2
x + p2y
) + ypx − xpy
− 1 − 2μ
r1− μ
r2− μ
r3(3)
and Hamilton equations of motion are
x = px + y,
y = py − x,
px = −py + (1 − 2μ)(x − x1)
r13/2
+ 8μ(x − x2)
r23/2
+ 8μ(x − x2)
r33/2
,
py = px + (1 − 2μ)y
r13/2
+ 4μ(2y − 1)
r23/2
+ 4μ(2y − 1)
r33/2
.
(4)
Let us remark that Eqs. (2) satisfy the following symme-try:
(t, x, y, x, y) −→ (−t, x,−y,−x, y).
3 Equilibrium points and their stability
In this section, we attempt to find the equilibrium pointsand examine their linear stability. The existence and thenumber of collinear and non-collinear equilibrium pointsof the problem depend on the mass parameter of the pri-maries.
It’s well known that the equilibrium points (x0, y0) arethe solutions of the equations
Ωx = 0, Ωy = 0
which yield
f1(x, y,μ) = x0 − (1 − 2μ)(x0 − x1)
[(x0 − x1)2 + y20 ]3/2
− μ(x0 − x2)
[(x0 − x2)2 + (y0 − y2)2]3/2
− μ(x0 − x3)
[(x0 − x3)2 + (y0 − y3)2]3/2= 0,
f2(x, y,μ) = y0 − (1 − 2μ)y0
[(x0 − x1)2 + y20 ]3/2
− μ(y0 − y2)
[(x0 − x2)2 + (y0 − y2)2]3/2
− μ(y0 − y3)
[(x0 − x3)2 + (y0 − y3)2]3/2= 0.
(5)
3.1 Locations of equilibrium points
The collinear points are the solutions of Eqs. (5) when y = 0.Clearly f2(x,0) = 0 for all values of x. Thus, the solutionsof f1(x,0) = 0 will correspond to equilibrium points on thex-axis, called collinear equilibrium points. The number ofsolutions of this equations depends on μ: in Fig. 3 f1(x,0)
is plotted for two different values of μ.The bifurcation value of μ for which the number
of zeros of f1(x,0) changes from 2 to 4 is μ∗ �0.2882761912857439. In Table 1 the equilibrium points, la-beled as in Baltagiannis and Papadakis (2011), dependingon μ are shown.
3 Page 4 of 11 Astrophys Space Sci (2015) 358:3
Fig. 3 Function f1(x,0) for the values μ = 0.25 (left) and μ = 0.45 (right)
Table 1 Number of collinearequilibrium points dependingon μ
μ Number Equilibrium points Position along the x-axis
(0,μ∗) 2 L2, L3
μ∗ 3 L2, L3, L1 = L4
(μ∗,0.5) 4 Li , i = 1, . . . ,4
The coordinates of the collinear points Li are (0, xLi),
i = 1, . . . ,4. In Fig. 4 the behavior of xLias a function of
μ is shown. When μ → 1/2, the equilibrium points L1 andL2 tend to the same point at the position of m1 which in thiscase vanishes.
The non-collinear points can be found by solving Eqs. (5)simultaneously when y �= 0, that is, f1(x, y,μ) = 0 andf2(x, y,μ) = 0.
Notice that
f1(x,−y,μ) = f1(x, y,μ) and
f2(x,−y,μ) = −f2(x, y,μ).
Thus, for each solution (x, y) of the equations with y > 0,(x,−y) is also a solution. Thus, the non-collinear equilib-rium points come in pairs.
The equations can be seen as the intersection of the zerolevel sets of equations f1 and f2. In Fig. 5 we show thissituation, in the half plane y > 0, for different values of μ.
Graphically, we observe 3 or 2 intersection points (al-though it seems there is a forth one, it corresponds to a sin-gularity). There is a value μ∗∗ � 0.44020149999 such that
Table 2 Non-collinear equilibrium points of the ERFBP dependingon the mass parameter. The bifurcation value is approximately μ∗∗ =0.44020149999
μ Number of equilibria Non-collinear points
(0,μ∗∗) 6 L5, L6, L7, L8, L9, L10
μ∗∗ 5 L5, L6, L7, L8, L9 = L10
(μ∗∗,0.5) 4 L5, L6, L7, L8
there exist 3 intersection points for μ < μ∗∗, otherwise thereare only 2 intersection points. In Fig. 6 we show the posi-tions of the equilibrium points L5,L7 and L9 varying μ.
The equilibrium points on the upper half plane are L5,L7 and L9, and the symmetric ones are, L6, L8 and L10
respectively. The summary of the number of non-collinearequilibrium points are shown in Table 2.
3.2 Stability of equilibrium points
In this section, we aim to study the stability of the equilib-rium points. To achieve this goal, we linearize the system (1)
Astrophys Space Sci (2015) 358:3 Page 5 of 11 3
around of each equilibrium point. By using numerical ap-proach, we attempt to find the eigenvalues of the matrix ofthe linearized system.
We recall that in the Hamiltonian case an equilibriumpoint is stable if the characteristic equation evaluated at theequilibrium point, has pure imaginary roots otherwise it isunstable.
If the linearization of (1) at the equilibrium point has acoefficient matrix A, so the linearized system is describedby
x = Ax, x = (x, y, x, y)T
where x is the state vector of the fourth body with respect tothe equilibrium points.
Fig. 4 Behavior of the coordinates xLi, i = 1, . . . ,4 as a function of μ.
The dotted line represents the x coordinate of the mass m1 = 1 − 2μ
Let (x0, y0) be the equilibrium point. It was shown thatat the collinear equilibrium points y = 0, and in this case thecoefficient matrix A is given by
A =
⎛⎜⎜⎝
0 0 1 00 0 0 1A1 0 0 20 A2 −2 0
⎞⎟⎟⎠
where
A1 = 1 − 2μ
( 14 + (x0 + 1
2
√3(1 − 2μ))2)3/2
+ 6(x0 + 12
√3(1 − 2μ))2μ
( 14 + (x0 + 1
2
√3(1 − 2μ))2)5/2
+ 2 − 4μ
(x0 − √3μ)3/2
and
Fig. 6 Positions of the equilibrium points L5,L7 and L9 varying μ.L9 ends for μ = μ∗∗ at L1. Stars denote the position of the equilibriumpoints for the initial value of μ = 0.0005
Fig. 5 Curves of the zero level sets of f1(x, y) (red) and f2(x, y) (blue) in the y > 0 half plane. Their intersections correspond to the triangularequilibrium points for μ = 0.2 (left), μ = 0.4 (center) and μ = 0.48 (right)
3 Page 6 of 11 Astrophys Space Sci (2015) 358:3
Fig. 7 Eigenvalues associated with L1 (left), L2 (center) and L4 (right) as a function of μ. For real eigenvalues, λ > 0 is plotted. For pure complexeigenvalues, only the positive imaginary part is plotted. For complex eigenvalues a + bi, both the positives real and imaginary parts are plotted
A2 = 1 + 3μ
( 14 + (x0 + 1
2
√3(1 − 2μ))2)5/2
− 2μ
( 14 + (x0 + 1
2
√3(1 − 2μ))2)3/2
+ 2μ − 1
(x0 − √3μ)3/2
.
The characteristic equation therefore becomes
λ4 + (4 − A1 − A2)λ2 + A1A2 = 0. (6)
Introducing Λ = λ2, we have
Λ2 + (4 − A1 − A2)Λ + A1A2 = 0,
the solution of which is
Λ1,2 = 1
2
(−4 + A1 + A2 ±√
(4 − A1 − A2)2 − 4A1A2).
Consequently the eigenvalues λ1 = −√Λ1, λ2 = √
Λ1,λ3 = −√
Λ2, and λ4 = √Λ2 depend on the value of the
mass parameter μ.When varying μ, the stability of the equilibrium points
changes. We denote by nr , ni and nc the number of real, purecomplex or complex with non-zero real part respectively.
• Equilibrium point L1: the number of eigenvalues of eachtype depends on the values of μ. In the next table we sum-marize the different scenarios: complex saddle, two sad-dles or a center × saddle.
μ nr ni nc
(μ∗,0.428337819] 0 0 4[0.42833782,0.44020157] 4 0 0[0.44020162,0.5) 2 2 0
In Fig. 7 the eigenvalues, as functions of μ are shown.• Equilibrium point L2: for all values of μ ∈ (0,0.5), nr =
ni = 2. Thus the equilibrium point is of type center ×saddle for all values of μ. See Fig. 7.
• Equilibrium point L3: for all values of μ ∈ (0,0.5),nr = 0, and for a small range of values of μ, all the eigen-values are pure imaginary, so the equilibrium point is sta-ble (see Fig. 8).
μ nr ni nc
(0,0.0027096302] 0 4 0[0.0027096305,0.5) 0 0 4
• Equilibrium point L4: for all values of μ ∈ (μ∗,0.5),nr = ni = 2. Thus the equilibrium point is of type cen-ter × saddle for all values of μ. See Fig. 7.
For the non-collinear equilibrium points, we suppose(x0, y0) are the coordinates of the point which results fromthe solution of Eqs. (5) with y �= 0. In the present case thecoefficient matrix A of the linearized system is given by
A =
⎛⎜⎜⎝
0 0 1 00 0 0 1
A11 A12 0 2A12 A22 −2 0
⎞⎟⎟⎠ (7)
where
A11 = 1 + 3(x0 − x1)(1 − 2μ)
((x0 − x1)2 + y2)5/2− 1 − 2μ
((x0 − x1)2 + y20)3/2
+ 3(x0 − x2)2μ
((x0 − x2)2 + (y0 − y2)2)5/2
− μ
((x0 − x2)2 + (y0 − y2)2)3/2
+ 3(x0 − x3)2μ
((x0 − x3)2 + (y0 − y3)2)5/2
− μ
((x0 − x3)2 + (y0 − y3)2)3/2,
A12 = 3(x0 − x1)(1 − 2μ)y0
((x0 − x1)2 + y20)5/2
+ 3(x0 − x2)(y0 − y2)μ
((x0 − x2)2 + (y0 − y2)2)5/2
+ 3(x0 − x3)(y0 − y3)μ
((x0 − x3)2 + (y0 − y3)2)5/2,
A22 = 1 + 3y20(1 − 2μ)
((x0 − x1)2 + y20)5/2
− 1 − 2μ
((x0 − x1)2 + y20)3/2
− μ
((x0 − x2)2 + (y0 − y2)2)3/2
Astrophys Space Sci (2015) 358:3 Page 7 of 11 3
Fig. 8 Eigenvalues associated with L3 as a function of μ. For pure complex eigenvalues, only the positive imaginary part is plotted (right plot).For complex eigenvalues a + bi, both the positives real and imaginary parts are plotted
+ 3(y0 − y2)2μ
((x0 − x2)2 + (y0 − y2)2)5/2
− μ
((x0 − x3)2 + (y0 − y3)2)3/2
+ 3(y0 − y3)2μ
((x0 − x2)2 + (y0 − y2)2)5/2.
Note that in this case the coefficients Aij depend on thethree variables (x0, y0,μ). In this case the characteristicequation associated with the matrix (7) reduces to the fol-lowing form:
λ4 + (4 − A11 − A22)λ2 + A11A22 − A2
12 = 0. (8)
As before, we introduce the variable Λ = λ2, obtaining theequation
Λ2 + (4 − A11 − A22)Λ + A11A22 − A12 = 0
and roots are given by
Λ1,2 = 1
2
(−4 + A11
+ A22 ±√
(4 − A11 − A22)2 − 4(A11A22 − A12))
with eigenvalues λ1,2 = ±√Λ1 and λ3,4 = ±√
Λ2.We solve numerically (8), and the μ dependent eigenval-
ues are summarized in the next table with the L5 and L7
eigenvalues plotted in the relevant ranges in Figs. 9 and 10:
L5,6 μ ∈ (0,0.018858526)
μ ∈ (0.018858526,0.5)
ni = 4nc = 4
L7,8 μ ∈ (0,0.5) nr = ni = 2L9,10 μ ∈ (0,μ∗∗) nr = ni = 2
We found that L3 when μ ∈ (0,0.00271096), and L5,6
when μ ∈ (0,0.01885853) possess two pairs of imaginaryeigenvalues ±iω1 and ±iω2. Then for these intervals of pa-rameter values the points, L3 and L5,6 are linearly stable.Here we shall examine the nonlinear stability of these ellip-tic points.
If the quadratic Hamiltonian is not positive definite, lin-ear stability does not imply nonlinear stability. Since this isthe case for our Hamiltonian, we have to pay attention tothe nonlinear stability of the equilibrium points L3 for μ ∈(0,0.00271096), and L5,6 for m2,3 ∈ (0,0.01885853] whichmay be determined using the Arnold–Moser Theorem.
Let us remark that an equilibrium point is spectrally sta-ble if all eigenvalues of its linearization are pure imaginary.Note, however, that spectral stability does not imply lin-ear stability. Here is where the integral of the motion rep-resented by the Hamiltonian function comes to our rescue,because if the normalized quadratic part is positive definitethen we can apply the stability theorem of Lyapunov and usethe Hamiltonian as a Lyapunov function. If it is not positivedefinite, then we must use the higher orders of the Birkhoffnormal form around the equilibrium.
4 Normal form
In the previous section we have seen that the linearizationsof the ERFBP at L3 when μ ∈ (0,0.002710), and L5,6 whenm2,3 ∈ (0,0.0188), have two pairs of pure imaginary eigen-values. Thus, we have already established linear stability.The next step is to determine the nonlinear stability, so weshould transform the Hamiltonian into its Birkhoff normalform.
The application of the Arnold–Moser theorem to the sta-bility of elliptic equilibria of Hamiltonian systems with twodegrees of freedom develops into two steps: diagonalizationof the quadratic part of the Hamiltonian and construction ofthe fourth order normal forms in the absence of resonancesof order up to four.
We translate the origin of the coordinate system to theequilibrium point Lk
x = x0 + q1, y = y0 + q2,
px = px0 + p1, py = py0 + qp2.
3 Page 8 of 11 Astrophys Space Sci (2015) 358:3
Fig. 9 Eigenvalues associated to L5 as a function of μ. For pure complex eigenvalues, only the positive imaginary part is plotted (right plot). Forcomplex eigenvalues a + bi, both the positive real and imaginary parts are plotted
Fig. 10 Eigenvalues associated to L7 and L9 as a function of μ. For real eigenvalues, λ > 0 is plotted. For pure complex eigenvalues, only thepositive imaginary part is plotted
In these new variables the equilibrium state is: qj = pj = 0,j = 1,2.
Then, we expand the Hamiltonian (3) in the vicinity of(0,0,0,0) in power series
H(q1, q2,p1,p2) = H2 + H3 + H4 + · · · + Hm. (9)
Forms Hm up to m = 4 which are necessary in the fu-ture analysis are too lengthy to be reproduced here, andfurthermore all the work will be done numerically. Notethat the eigenvalues will depend on the equilibrium point(x0, y0,px0,py0), which is a function of μ, and cannot bealgebraically calculated. It is known that if H2 is a sign defi-nite function, then the equilibrium point is stable. However,for all values in the grid we have examined the function H2,and found that it is not sign definite; hence, we cannot drawany conclusion about the whole system stability in terms of
its linear approximation. Normalization over the reals takesH2 to the following form:
H2(Q1,Q2,P1,P2) = ω1
2
(Q2
1 + P 21
) ± ω2
2
(Q2
2 + P 22
)(10)
where ω1 > ω2.Two situations must be considered. On the one hand, for
the plus sign, according to the classical Lyapunov theory,a result of Dirichlet ensures the stability of the origin forthe whole system defined by (3), because H2 is positive-defined (see Siegel and Moser 1971). On the other hand, forthe minus sign in H2 it is not sign-defined, and Dirichlet’stheorem of stability cannot be applied. However, for thisinteresting case, Arnold–Moser’s theorem (Arnold 1964;Markeev 1971) gives sufficient conditions to determine thestability character of the origin if the fundamental frequen-cies ω1 and ω2 satisfy a general condition of irrationalityand the Hamiltonian is in Birkhoff normal form. Since thesystem is intractable analytically the Birkhoff normal form
Astrophys Space Sci (2015) 358:3 Page 9 of 11 3
is obtained numerically and the irrational condition deter-mined a posteriori. In Alvarez-Ramírez et al. (2014) we dis-cuss with some detail the procedure to obtain a Birkhoff nor-mal form, and more details can be found in Simó (1989) andStuchi (2002). The real normal form up to order four withthe minus sign in H2 is given by:
Fig. 11 Graph D4 versus μ. The region where D4 has an anomalousbehavior corresponds to Fig. 12 where μ-axis has been enlarged and itcorresponds to the 2 : 1 resonance
H4(Q1,Q2,P1,P2) = ω1
2
(Q2
1 + P 21
) − ω2
2
(Q2
2 + P 22
)
+ c4000(Q2
1 + P 21
)2
+ c2200(Q2
1 + P 21
)(Q2
2 + P 22
)+ c0400
(Q2
2 + P 22
)2 + O(5) (11)
where Q2i + P 2
i , i = 1,2, are action variables; the coeffi-cients c4000, c2200 and c0400 are numerical coefficients de-termined for each set (μ,ω1,ω2).
More explicitly, the theorem is as follows.If the Hamiltonian function is such that
(1) the characteristic equation of the linear system has twoimaginary eigenvalues ±Iω1 and ±Iω2;
(2) n1ω2 + n2ω2 �= 0, where n1 and n2 are integers suchthat 0 < |n1| + |n2| �= 4;
(3) D4 = ω22c4000 + ω1ω2c2200 + ω2
1c0400 �= 0, then theequilibrium position is Lyapunov stable.
4.1 Computation of D4 for L3
In order to use the Arnold–Moser theorem, one has to studythe function D4(μ) expressed in item (3) of Arnold–Mosertheorem. With a fine grid of values of μ ∈ (0,0.00271096)
we calculate the equilibrium point, the eigen-system, nor-malize the Hamiltonian to form (11) and finally evaluate D4.The behavior of D4 with μ is shown in Figs. 11 and 12.
Examining Fig. 11 we noted that there is a 2 : 1 resonancewhich was found with numerical refinement. This resonanceoccurs at the value of μ = 0.0017577. In Fig. 12 it is clear
Fig. 12 Graph D2 versus μ. The region where D2 crosses the μ-axis has been enlarged and it corresponds to the 2 : 1 resonance (see Fig. 11)
3 Page 10 of 11 Astrophys Space Sci (2015) 358:3
Fig. 13 Graph D4 versus μ. The regions where D4 �= 0 correspond tovalues of μ for which the equilibrium point is stable. The value of μ
where the determinant diverges is a 2 : 1 resonance. The value of μ forwhich D4 = 0 requires a higher order normal form
that D4 does not change sign except at the resonance, im-plying that L3 is stable, except at the resonance which hasto be studied separately.
4.2 Stability of the symmetrical equilibrium points L5
and L6
Again, in order to apply the Arnold–Moser theorem, one hasto check whether the determinant
D4 = c4000ω22 + c2200ω1ω2 + c0400ω
21 (12)
evaluated for μ ∈ (0,0.018858526) does not vanish. Us-ing the same procedure as in the case of L3, we deter-mine numerically the values of D4 in the interval of μ
values for which L5,6 exist. The result is represented inFig. 13.
As can be seen in Fig. 13, D4 = 0 at μ =0.00580900433 . . ., and its stability can be checked witha sixth order normal form which will be dealt with inSect. 6. Apart from this value of μ and the resonant valueμ = 0.01208744, the interval μ ∈ (0,0.18858526) is stable.
5 Stability at the 2 : 1 resonance for L3 andsymmetrical points L5 and L6
We follow Markeev (1971) to discuss the stability in thecase of 2 : 1 resonance. Suppose the Hamiltonian hasbeen expanded in Taylor series and expressed in terms of
variables which take the quadratic part to its real normalform:
H = 1
2
(p2
1 + ω21q
21
) − 1
2
(p2
2 + ω22q
22
)+
∑ν1+ν2+μ1+μ2=3
hν1ν2μ1μ2qν11 q
ν22 p
μ11 p
μ22 . (13)
In order to take the Hamiltonian in a form suitable forBirkhoff transformation it is convenient to complexify (13)through the transformation:
q1 = 1
2q ′
1 + I
ω1p′
1, p1 = 1
2Iω1q
′1 + p′
1,
q2 = −1
2q ′
2 + I
ω2p′
2, p2 = −1
2Iω2q
′1 + p′
2.
In the new variables the Hamiltonian takes the form:
H = Iω1q′1p
′1 + Iω2q
′2p
′2
+∑
ν1+ν2+μ1+μ2=3
h′ν1ν2μ1μ2
q ′ν11 q ′
2ν2p′μ1
1 p′μ22 (14)
where
h′ν1ν2μ1μ2
= xν1ν2μ1μ2 + Iyν1ν2μ1μ2
h′ν1ν2μ1μ2
= xν1ν2μ1μ2 + Iyν1ν2μ1μ2
(−ω1
2
)(ν1−μ1)
×(
ω2
2
)(ν2−μ2)
and the xν1ν2μ1μ2 and yν1ν2μ1μ2 depend on the coefficientsof Hamiltonian (13) (see Markeev 1971 for more details).
Now, by making a Birkhoff transformation q ′p′ → QP ,we obtain the third order normal. As expected the resonantmonomials Q1P
22 and P1Q
22 cannot be eliminated, so that
the third order normal form is as follows:
H = Iω1Q1P1 + Iω2Q2P2 + h′1002Q1P
22 + h′
0210P1Q22
+ O(|Q|4). (15)
After a canonical transformation which turns the Hamilto-nian real, Markeev introduces cylindrical variables
qj = √rj sin(φj − θj ), pj = √
rj cos(φj − θj ) (16)
with θ2 = 0, sin(θ1) = y1002/
√x2
10002 + y21002 and cos(θ1) =
x1002/
√x2
10002 + y21002. This change takes the third order
resonant normal form to the following form:
H = 2ω2r1 − ω2r1
−√
ω2(x2
1002 + y21002
)r2
√r1 sin(φ1 + 2φ2)
+ H (rj , φj ), (17)
H is 2π -periodic in φj and of order (r1 + r2)2.
Based on the normal form (17) the following theorem isproven in Markeev (1971):
Astrophys Space Sci (2015) 358:3 Page 11 of 11 3
If the Hamiltonian of the perturbed motion is such thatx2
1002 + y21002 �= 0 then the equilibrium position is unsta-
ble. If x21002 + y2
1002 = 0 and D4 = ω21c4000 + ω1ω2c2200 +
ω22c0400 �= 0 the equilibrium point is Lyapunov stable.
5.1 Application to L3 and the symmetrical pair L5 andL6
We found the following third order normal form for the equi-librium point L3:
HNL3 = 0.88595523I Q1P1 − 0.44297762I Q2P2
+ 0.88196400(1 − I )(Q1P
22 + P1Q
22
)+ O
(|Q|4) (18)
where |Q| =√
Q21 + Q2
2 + P 21 + P 2
2 and (Q1,Q2,P1,P2)
are the complex variables which diagonalize the originalHamiltonian over the complex field.
Clearly the square of the coefficient of Q1P22 is not zero
and so the equilibrium point is unstable at the 2 : 1 resonanceat the L3 equilibrium point.
Analogously, we now apply the same theorem for the2 : 1 resonance occurring at the symmetrical points, L5 andL6. The third order normal form is as follows:
HN3L6 = 0.88954051I − 0.44477026I
+ (−3.4664817 + 3.5900172I )Q1P22
+ (−3.5900172 + 3.4664817I )P1Q22 + (|Q|4)
where |Q| is as above. As in the case of L3 the sum of thesquares of the Q1P
22 coefficient is not zero and the 2 : 1
resonant equilibrium points L5 and L6 are also unstable. Werecall that these points are symmetrical, so it is sufficient toshow the instability of one of them.
6 Nonlinear stability for the equilibrium L5 andL6 for μ = 0.580900433 . . .
This section is devoted to the investigation of nonlinear sta-bility of L5 and L6 at μ = 0.580900433 . . . for which thevalue of D4 is zero. In this case the normal form has to betaken to order six (or higher), as it was the case of the stabil-ity of L4 and L5 in the restricted three body problem solvedby Meyer et al. (2009).
As it is well known, to evaluate the Birkhoff normal formby the Lie series one has to solve the following homologicalequation for each order k:
LGkH2 + Zk = Fk with F3 = H3 (19)
where Fk is a combination of Poisson brackets of order k
involving the generating functions G of order less than k.
We note that the normal form Zk has to be determined at thesame time as Gk . Again using Maple we found the followingsixth order normal form:
HN6 = 24.408716J 31 − 403.4904016J 3
2 − 237.59255J 21 J 1
2
+ 1308.0177J1J22 (20)
where Ji = Q2i +P 2
i , i = 1,2, are the action variables. Withthe values ω1 = 0.954668729I and ω2 = 0.28874226I wefind that
D6 = −25.17642217 (21)
and conclude that L5 and L6 at μ = 0.580900433 . . . arenonlinearly stable.
7 Conclusion
The neighborhood of L3 for μ ∈ (0,0.00271096) is sta-ble except for μ = 0.00017577, where we have an unstable2 : 1 resonance. In the case of the symmetrical non-collinearequilibria, L5 and L6 in the interval μ ∈ (0,0.018858526),the equilibrium point is stable except at the 2 : 1 resonanceat μ = 0.01208744. For μ = 0.580900433 . . . , where D4 iszero, but D6 �= 0 and L5 and L6 for this value is nonlinearlystable.
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