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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, pp. 545-557. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231211
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70M20
Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem
B. S.Bardin, E. A. Sukhov, E. V. Volkov
We consider the planar circular restricted four-body problem with a small body of negligible mass moving in the Newtonian gravitational field of three primary bodies, which form a stable Lagrangian triangle. The small body moves in the same plane with the primaries. We assume that two of the primaries have equal masses. In this case the small body has three relative equilibrium positions located on the central bisector of the Lagrangian triangle.
In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.
Keywords: Hamiltonian mechanics, four-body problem, equal masses, periodic motions, orbital stability, symplectic mapping, nonlinear analysis, numerical computation
Received November 20, 2023 Accepted December 05, 2023
This work was carried out at the Moscow Aviation Institute (National Research University) and was supported by the grant of the Russian Science Foundation (project Nr. 22-21-00729).
Boris S. Bardin bsbardin@yandex.ru Evgeniy V. Volkov evvolkov94@mail.ru
Moscow Aviation Institute (National Research University) Volokolamskoye sh. 4, Moscow, 125080 Russia
Mechanical Engineering Research Institute of the Russian Academy of Sciences M. Kharitonyevskiy per. 4, Moscow, 101990 Russia
Egor A. Sukhov sukhov.george@gmail .com
Moscow Aviation Institute (National Research University) Volokolamskoye sh. 4, Moscow, 125080 Russia
1. Introduction
The restricted four-body problem implies the motion of a small body with infinitesimal mass in the Newtonian gravitational field of three primaries with finite masses. This problem has been a subject for extensive analytical and numerical research and continues to draw significant attention from the scientific community both due to theoretical considerations and also due to possible applications in astronomy and space exploration. In this work we consider the important special case known as the planar circular restricted four-body problem in which the primaries form a stable Lagrangian triangle and the small body moves in the same plane with the primaries. Among the topics of particular interest for this problem are the relative equilibria and periodic orbits. A notable class of these equilibria exists when the bodies constitute the so-called central configurations. Central configurations are characterized by the net attracting force acting on the small body being directed towards the system's barycenter. The resulting equilibria are similar to the libration points in the restricted three-body problem. The small body can also move in periodic orbits about the equilibria associated with central configurations. These periodic motions constitute the so-called natural families defined by the problem's parameters. Natural families can be obtained analytically in the small neighborhoods of the equilibria and can also be continued and investigated outside of these small neighborhoods using numerical methods. Existence and orbital stability of these periodic orbits bears special importance for analyzing the dynamics of the problem.
The periodic orbits in the four-body problem have been studied in many works, including [1-7]. Notably, the works [1-4] address the case of two primary bodies with equal masses and present an investigation of symmetric and nonsymmetric planar periodic orbits for particular values of the mass ratio parameter. In [5] the areas of permitted motion and the existence of periodic motions for a planar four-body problem are studied. The work [6] deals with local equilibria in the case of collinear configuration of the primary bodies and presents a numerical study of the periodic motions emanating from the said equilibria. In [7] the periodic motions in the restricted four-body problem are constructed and analyzed using spatial halo-type solutions of the restricted circular three-body problem as first-order approximations. The linear stability analysis of periodic motions emanating from stable central configurations was carried out in [8].
In recent years new and effective methods have been developed for the rigorous nonlinear stability analysis of periodic motions in Hamiltonian systems. All these methods are based on normalizing the equations of perturbed motion and applying KAM theory. An important first step of such rigorous analysis involves reducing the initial problem by introducing local coordinates in the neighborhood of the periodic orbit and obtaining the equations of perturbed motion. Even though general theory shows that it is always possible to introduce such local coordinates, obtaining a canonical transformation that introduces them is a nontrivial task. There are a number of works dealing with this problem which present methods for obtaining the aforementioned canonical transformation. The most universal method has been recently proposed in [9]. It allows one to introduce local coordinates both for periodic motions constructed analytically and for those obtained numerically. The second step deals with analyzing the stability in the reduced problem. An effective rigorous approach to this problem is proposed in the work [10] where a symplectic map generated by the phase flow of the reduced system is constructed and the stability of its stationary point is considered. These methods can be applied for both the periodic motions obtained analytically and for those obtained numerically as sets of corresponding initial values. This bears critical importance in the scope of this work since the periodic orbits studied herein can only be obtained through a numerical continuation procedure.
In this work we propose a numerical procedure implementing the aforementioned methods and apply it to investigate the periodic orbits emanating from stable relative equilibria associated with the central configurations in the planar circular restricted four-body problem. We consider the nonresonant periodic orbits as well as resonant ones corresponding to the resonances of the third and fourth order. The resonances of the first and second orders are not considered as they require special treatment that is beyond the scope of this work. In Section 2 we write out the canonical equations of motion for the small-mass body and specify the stable central configurations with the associated relative equilibria. We also specify the families of periodic motions emanating from these equilibria. In Section 3 we reduce the initial problem by introducing local coordinates in the neighborhood of a known periodic orbit and performing an isoenergetic reduction on the energy level of this orbit. In Section 4 we construct a symplectic transformation and investigate the stability of its stationary point to draw conclusions on the orbital stability of the considered periodic orbit. In Section 5 we implement the method described in Sections 3, 4 as a numerical procedure and apply it for analyzing the orbital stability of periodic orbits emanating from the stable relative equilibria associated with central configurations for all admissible values of the problem's parameters.
2. Problem formulation
We consider the planar circular restricted four-body problem with Pi (i = 1, 2, 3) being the primary bodies, or primaries with masses mi (i = 1, 2, 3), respectively, and P being a small body with infinitesimal mass. The primaries move in circular orbits about the system's barycenter with the small body moving in the same plane with the primaries. We assume the masses m2 and m3 of the two primaries P2 and P3 to be equal and introduce a mass ratio parameter ¿t = m ^m [H]-The primaries Pi (i = 1, 2, 3) form an equilateral Lagrangian triangle shown in Fig. 1 [12]. We only consider the case when the triangle P\P2P3 is stable. This takes place when the following condition is fulfilled [13]:
(2-1)
A7?
\
ïA
/
\
\
O \
\
-►
Ps Î
Fig. 1. Dimensionless synodic coordinates for the restricted planar circular four-body problem. P1, P2 and P3 are the primaries, while P is the small-mass body
To study the relative motion of the small body P, we introduce the right-handed coordinate system Oxyz with its origin O located at the center of the side P2P3 of the Lagrangian triangle P\P2P3• The axis Oy passes through the body P1 and the axis Oz is perpendicular to the
plane of motion. We then apply the following coordinate transformation:
x = y = rn,
to introduce dimensionless coordinates n where r is the distance between the primaries.
The equations of motion of the small body P can be written in the following canonical form [14]:
dv dp^ dv dp^ dv d(
V3,
H = ô (Pi+Pï) P " ^ri1 " ~Pvt ~
dv
1 - 2 n Pi
dH dn '
f f
P'2 Pi
(2.2)
where
Pi =
« + "-x
P2 =
1/2
+ n2
P3 =
1/2
+ n2
The independent variable v = ut can be regarded as the angle of rotation of the Lagrangian triangle formed by the primaries, where u2 = f (ml+2m2)r-3 and f is the gravitational constant. The system (2.2) has the following steady-state solution:
£ = 0, r] = ri„ pç = + —(1 - 2fi), pv = 0,
(2.3)
where n* is a real root of the following equation:
r? - ^-(1 - 2fi) -
4(1 -2(jl) sign (2r?- v/3) 16fir]
(2rj - V?>Y
(4n2 + 1)3/2
0.
The solution (2.3) describes three relative equilibrium points A, B, C located on the axis On (Fig. 2). In this case the small body P and the primaries P.\ constitute the so-called central configurations. These central configurations are deltoid-shaped and rotate at constant angular velocity w in the absolute space. It has been shown [14, 15] that points A and B are unstable for all values of f, while point C is unstable for f > f2 œ 0.00270963 and it is linearly stable for 0 < f < f2.
According to Lyapunov's theory [16], in the small neighborhood of a stable relative equilibrium of a Hamiltonian system there exist two types of periodic solutions categorized by their period: short-periodic solutions with period Ts = ^ and long-periodic motions with period TL =
= (TL > Ts). Here Qs and QL are frequencies of the Hamiltonian system linearized in the neighborhood of the considered equilibrium.
In our case Lyapunov's solutions represent periodic orbits around central configurations. Periodic orbits form the so-called natural families defined by the problem's parameters, including the energy constant h.
In [8], families of both short- and long-periodic orbits were constructed analytically in the small neighborhood of the stable equilibrium point C. The short-periodic orbits were also studied numerically for all admissible values of the problem's parameters f and h. Figure 3 shows the
2
2
A
i A
\
\
\
\
\
\
>B x O
c
\
p.a i
Fig. 2. Central configurations and the associated relative equilibria A, B and C of the small-mass body P in the restricted planar circular four-body problem with bodies P2 and P3 having equal masses
Fig. 3. Existence domain for the family of short-periodic motions emanating from the relative equilibrium C
existence domain of the short-periodic orbits in the problem's parameter space. The considered orbits exist for ¡> 0 between the curve defined by the energy level of the equilibrium C and the curve S. Figure 4 shows the shapes of several short-periodic orbits computed for different values of the energy constant h and fixed value of the mass ratio parameter ¡.
The period of the considered orbits depends continuously on the problem's initial values, which leads to their Lyapunov instability. However, they can be orbitally stable, i.e., stable with respect to normal variations of the periodic orbit in the system's phase space.
In this work we investigate the nonlinear orbital stability of short-periodic motions emanating from the stable equilibrium point C of the restricted planar circular four-body problem.
3. Local variables and isoenergetic reduction
In this section we describe a method for introducing local variables in the neighborhood of the considered periodic orbits. We also describe an isoenergetic reduction algorithm that allows
Fig. 4. Left: Orbits of the short-periodic motions emanating from the relative equilibrium C for j = 0.001. The orbits S1, S2 and S3 correspond to the energy constant values of —1.4999, —1.4955 and —1.4938, respectively. The orbit S2 belongs to the region of orbital instability. Right: orbits S4, S3 and S5 situated in the neighborhood of upper boundary S of the existence domain (Fig. 3) with the values of j = 0.0002, j = 0.001 and j = 0.002, respectively
conclusions on the orbital stability of periodic orbits by investigating a simpler problem of the equilibrium stability of a reduced system.
We assume that the initial Hamiltonian system (2.2) possesses a known T-periodic solution
£ = fi(v), n = f2(v), Pc = 9i(v) Pv = 92(v)
(3.1)
that describes a periodic orbit of the small body P. To investigate the motion in the neighborhood of the unperturbed periodic orbit, we introduce local variables by means of the canonical transformation [9]
n P iPn.
= z(t, u, p, v) =
fi(v)' /2 (v) 9i(v) 92 (v)
+ uZu(v) + vZ„(v) + [p + G(v, u, p, v)]Zra(v)
(3.2)
that has the following properties:
1. In the new variables the periodic solution (3.1) takes the form u = p = v = 0.
2. Z is a T-periodic vector-function of v.
3. Z is an analytical function of u, p, v in the small neighborhood of the periodic motion. The vector functions Zu, Zv, Zn in (3.2) are defined as follows:
L(dA
V\dv'
4fi _(kh (kh
dv} dv ' dv
z, = — f-^h dfa <H'2 \
v vydv1 dv' dv' dv)
z = — (^1 -^Ei (Mi jMi
n V V dv ' dv' dv' dv
T
T
T
(3.3)
(3.4)
(3.5)
u
dfiY , (df2\\ (d9i\\ fdg2^2
where V is the phase velocity:
v=№) +{it) Hi^J H^J- (3-6)
The function G(v, u, p, v) in (3.2) can be written as
G = G2 + G3 + G4 + O5, (3.7)
where
C2 = ^ [A (.t2 + y2) F2 - 2r](B + C)F - V], (3.8)
G
G3 = -^[Arl + (Bx + Cy)V], (3.9)
G
= ^ [(A2 (x2 + y2) - 4(Bx + Cy)2) F2 - lOA^.r + Cy)V - 5A2r?2], (3.10)
^ = (kh(iJi + (khdHi _ _ (3ii)
dv dv2 dv dv2 dv dv2 dv dv2 '
B = ^id2^ + (kh(Pjh _ _ (kh(Pjh /3 12)
dz/ dz/2 dz/ dz/2 dz/ dz/2 dv dv2 ' c = + _ (Ml(ijtL _ iil^Jl (313)
dv dv2 dv dv2 dv dv2 dv dv2 It has been shown in [9] that G is an analytical function of the variables u, p, v in the small neighborhood of the periodic orbit and it is also T-periodic in v.
On applying the transformation (3.2) to the initial system given by (2.2) we obtain the new Hamiltonian as an expansion into power series of the new variables u, p, v:
r = r2 + r3 + r4 +r, (3.14)
where
r2 = P + P2(v, U, v),
r = P3(v, u, v) + p^i(v, u, v), (3.15)
r4 = p2x(v) + p^2(v, u, v) + ^(v, u, v).
In (3.15) r are terms of the fifth order and higher, ^i (i = 1, 2) and Pj (j = 2, 3, 4) are forms of orders i and j whose coefficients depend periodically on v. It can be also deduced from the structure of r2 that the variable p has second order of smallness as opposed to the variables u, v which have first order of smallness.
In order to analyze the orbital stability of the periodic orbits (3.1), we perform an isoen-ergetic reduction. That is, we investigate the motion of the canonical system with Hamiltonian (3.14) on an isoenergetic level r = 0 which brings us to the reduced canonical system with one degree of freedom and the new Hamiltonian K represented by the following series:
K = K2 + K3 + K4 + ••• (3.16)
where Kk is a form of the kth degree in u, v. The forms Kk are periodic functions of v and the first three forms take the following explicit form:
K2 = ^
K3 = P3 - ^2, (3.17)
K4 = Xtl + - ^2] P2 - P3^1 + ¥4.
The forms p. in (3.17) are the same as in (3.15).
4. Orbital stability analysis
The orbital stability problem for periodic orbits of the canonical system with (3.16) is equivalent to the Lyapunov stability problem of an equilibrium u = v = 0 [9].
In particular, the linear orbital stability analysis of the periodic orbit (3.1) can be performed by investigating the stability of the equilibrium point u = v = 0 of the linear system with Hamiltonian K2:
du d K2 dv d K2 dv
dv
dv
du
(4.1)
To accomplish this we consider the characteristic equation
A2 - 2aA + 1 = 0,
(4.2)
where a = ^[a;11(T) + x22(T)] and a;11(z/), x22(u) are elements of the matrizant X(z/) of the system (4.1).
Lyapunov's general theory of stability gives the following stability and instability conditions for the considered equilibrium [16, 17]:
1. If |a| < 1, then the orbit is linearly orbitally stable.
2. If |a| > 1, then the orbit is orbitally unstable.
0.0005 0.0010 0.0015 0.0020 M
Fig. 5. Linear orbital stability diagram for the family of short-periodic motions emanating from the relative equilibrium C. The periodic orbits are linearly orbitally stable in the subdomains D1 and D2 and orbitally unstable in the subdomain D3
The above conditions also give us criteria for the linear orbital stability of the periodic orbits. For the short-periodic orbits emanating from the stable equilibrium point C we checked these conditions by solving the linearized system (4.1) numerically for the matrizant X(v) and calculating the coefficient a of the characteristic equation (4.2). As a result, we constructed the linear orbital stability domain of the considered orbits in the problem's parameter space n, h (Fig. 5). The short-periodic orbits are linearly orbitally stable in the subdomains D1 and D2 and are orbitally unstable in the subdomain D3. These results are in full agreement with the conclusions obtained earlier in [8].
To obtain rigorous conclusions regarding stability in the case of |a| ^ 1, nonlinear analysis is required. In this work we perform the nonlinear stability analysis using the approach described in [10]. This approach is based on reducing the stability problem for the nonlinear canonical system to analyzing the stability of a stationary point of a symplectic map generated by the phase flow of the system with Hamiltonian (3.16). Below we present a brief description of this approach.
First we introduce canonical variables
u = n11Q + n12P, v = n21Q + n22P, where the coefficients n^ in (4.3) are then calculated as follows:
(4.3)
n11 = -ÔKV12, n12 = °
«21 = Sn[yn - a], n22 = -kV 1 - a2,
1/2
(4.4)
y12Vl~a2 , ¿ = sign y12y/l-a2
where yij = xij(T) and xij are the elements of the matrizant X. This brings the Hamilto-nian (3.16) to the following form:
K*(Q, P, v) = K*2(Q, P, v) + K*3(Q, P, v) + K*4(Q, P, v) + ••• (4.5)
The canonical system with Hamiltonian (4.5) generates the following symplectic transformation
dFM
Q1 = G
P1
O dF30 dF30
Vo ap '
dP0
P I 9F30 + dQ0
dP0dQ0 dP0
dPn
+ O4
d2F30 dF30 I dF40 I n
4
(4.6)
where Q0, P0 are the initial values of the variables Q, P for v = 0, and Q1, P1 are the values of the same variables corresponding to v = T. Transformation (4.6) is area-preserving and possesses a stationary point Q0 = P0 = 0. By virtue of the normalizing substitution (4.3) the linear part of this transformation denoted by the matrix G assumes the following simple form:
G
cos(2^0) sin(2n6 — sin(2n0) cos(2né
cos(2n0) = a,
where a is the coefficient of the characteristic equation (4.2).
The forms F3 = $3(Q0, P0, T) and F4 = $4(Q0, P0, T) in (4.6) are calculated using the following set of differential equations:
9<fr3(Q0, P0, v) dv
9<&4(Qo, P0, v) dv
= —r4(Q0, P0, v) —
= —r3(Q0, P0, v),
dr3(Q0,P0, v) d$3(Q0,P0, v)
(4.7)
dPn
dQn
where the forms and are obtained by substituting a transformation
X * (v )
Q0 P
K
(4.8)
into the terms K| and of the Hamiltonian (4.5). Here X*(v) is the matrizant of a linear canonical system with Hamiltonian K2,(Q, P).
To draw conclusions on the stability of the stationary point of the symplectic map (4.6), we introduce the following notation:
al = f30 — fl2, a2 = f 12 + 3f30, a3 = f22 — f40 — f04, bl = f21 — f03, b2 = f21 + 3f03, b3 = fl3 — f3l, k = 8(3f40 + f22 + 3f04) + 6(al b2 — a2 bl) — 8a2^2+
+9cot(3^a) (a2 + bl) + 3cot(^a) (a2 + b2), kl = 2[4a3 + 9albl — a2 b2 + 3cot(^a)(al a2 — blb2)], k2 = 8b3 — 9 (a2 — bl) + (a2 — b2) + 6cot(^a)(alb2 + a2bl),
i Sarccos(a)
where a = —27r .
The values fij in (4.8) are coefficients of the forms F3, F4 which can be represented as follows:
Fk = E fijQ0P0 (k = 3, 4).
i+j = k
The stability of the stationary point Q0 = P0 = 0 of symplectic map (4.6), i.e., the orbital stability of the corresponding periodic orbit (3.1) is determined via the following conditions [10, 18]:
1. In a nonresonant case where a / —^ and a / 0 the periodic orbit is orbitally stable if k / 0.
2. In the case of the third-order resonance where a = — | the orbit is orbitally unstable if at least one of the values al and bl is nonzero.
3. In the case of the fourth-order resonance where a = 0 the orbit is orbitally stable if \k\ >
> sjk'l + fc2 and orbitally unstable if \k\ < s/kf + fc2.
5. Numerical computation of orbital stability diagrams
Since there is no analytical representation for the considered periodic orbits, we have constructed a numerical procedure for rigorous analysis of their orbital stability. This procedure involves four steps:
1. Obtain initial values of the periodic orbits for given values of the problem's parameters (the mass ratio parameter ¡i and the energy constant h) using the numerical continuation method [19-21].
2. Determine the matrizant X and the values (4.4) by solving numerically the system of differential equations (2.2) and (4.1).
3. Determine the values (4.8) by solving the system (2.2), (4.1), (4.7).
4. Use the values (4.8) to check the stability conditions given at the end of Section 4.
-1.480
-1.482
-1.484
-1.486
-1.488
h -1.490
-1.492
-1.494
-1.496
-1.498
Unstable a = — \ .....Unstable a = 0--Stable a = 0 -к = 0
Fig. 6. Orbital stability diagram for the family of short-periodic motions emanating from the relative equilibrium C. The dash-dot curves correspond to the unstable resonant orbits in the case of third-order resonance (a = — The dotted curve corresponds to unstable resonant orbits in the case of fourth-order resonance (a = 0). The dashed curve corresponds to orbitally stable resonant orbits in the case of fourth-order resonance (a = 0). The solid curves correspond to the degeneracy case (k = 0)
Figure 6 shows the results of nonlinear stability analysis of short-periodic motions emanating from the stable equilibrium point C for all admissible values of the problem's parameters f and h. The dash-dot curves represent the case of third-order resonance when a = 0. In the case of third-order resonance the periodic orbits are orbitally unstable. The dotted and dashed curves represent the case of fourth-order resonance when a = — The dotted curve corresponds to the orbitally unstable resonant orbits and the dashed curve corresponds to the orbitally stable ones. The change in orbital stability occurs at the point P with coordinates f = 0.000536, h = -1.498403.
There also exist degeneracy curves corresponding to the case of k = 0 when additional analysis involving the terms of higher orders is required. In Fig. 6 these curves are shown as solid lines.
Conclusion
In this work we have performed a nonlinear orbital stability analysis of periodic orbits emanating from stable equilibria associated with central configurations in the restricted circular four-body problem. The orbital stability criteria were obtained using novel methods based on introducing local variables in the neighborhood of a periodic orbit and then constructing a sym-plectic mapping associated with it. The analysis was performed for all admissible values of the problem's parameters. Resonant periodic orbits with resonances of third and fourth order, as well as the nonresonant ones, were investigated.
Conflict of interest
The authors declare that they have no conflict of interest.
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