CC BY
0Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 563-576. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221201
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70M20
Numerical Orbital Stability Analysis of Nonresonant Periodic Motions in the Planar Restricted Four-Body
Problem
E.A. Sukhov, E. V. Volkov
We address the planar restricted four-body problem with a small body of negligible mass moving in the Newtonian gravitational field of three primary bodies with nonnegligible masses. We assume that two of the primaries have equal masses and that all primary bodies move in circular orbits forming a Lagrangian equilateral triangular configuration. This configuration admits relative equilibria for the small body analogous to the libration points in the three-body problem. We consider the equilibrium points located on the perpendicular bisector of the Lagrangian triangle in which case the bodies constitute the so-called central configurations. Using the method of normal forms, we analytically obtain families of periodic motions emanating from the stable relative equilibria in a nonresonant case and continue them numerically to the borders of their existence domains. Using a numerical method, we investigate the orbital stability of the aforementioned periodic motions and represent the conclusions as stability diagrams in the problem's parameter space.
Keywords: Hamiltonian mechanics, four-body problem, periodic motions, orbital stability
Received October 25, 2022 Accepted November 29, 2022
This work was supported by the grant of the Russian Science Foundation (project No. 22-21-00729) at the Moscow Aviation Institute (National Research University).
Egor A. Sukhov sukhov.george@gmail .com
Moscow Aviation Institute Volokolamskoye sh. 4, Moscow, 125080 Russia
Evgeniy V. Volkov evvolkov94@mail.ru
Moscow Aviation Institute
Volokolamskoye ave. 4, Moscow, 125080 Russia
Mechanical Engineering Research Institute of the Russian Academy of Sciences, M. Kharitonyevskiy per. 4, Moscow, 101990, Russia
1. Introduction
The problem of a small body with negligible mass moving in the Newtonian gravitational field of three primary bodies with nonnegligible masses is known as the restricted four-body problem. It has been thoroughly studied in many works and continues to attract considerable interest of the scientific community. This interest has been strengthened lately by commissioning of the new space telescopes that allow direct observation of exo-solar planetary systems and create a potential for discovering configurations previously considered only in theory. Among the topics of interest are relative equilibria and periodic orbits in the planar restricted four-body problem. An important class of these equilibria appears when the bodies assume the so-called central configurations. Central configurations are characterized by the net attracting force acting on the small-mass body being directed towards the system's center of mass. The resulting equilibria are analogous to the libration points in the classical three-body problem. The small body can also move in a periodic orbit about the equilibrium associated with a central configuration. Periodic orbits form the so-called natural families that can be constructed analytically in a small neighborhood of a relative equilibrium. These families can also be continued and investigated outside of small neighborhoods of the equilibria using numerical continuation methods.
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 these 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.
In this work we consider periodic motions emanating from stable relative equilibria associated with the central configurations in the planar circular restricted four-body problem with two of the primary bodies having equal masses. 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. In Sections 3 and 4 we describe the numerical continuation algorithm used to construct the periodic motions outside of small neighborhoods of the relative equilibria and investigate their linear orbital stability. In Section 5 we obtain analytical expressions for the periodic motions in small neighborhood of the relative equilibria using the method of normal forms. In Section 6 the numerical analysis of the periodic motions is performed. We continue these motions numerically to the borders of their existence domain and construct a linear orbital stability diagram in the problem's parameter plane. We also present typical orbital plots for the periodic motions and verify the results using Poincare maps.
2. Formulation of the problem
We consider the planar restricted four-body problem where P\ (i = 1, 2, 3) are primary bodies with masses m. (i = 1, 2, 3), respectively, and P is a small body with negligible mass that moves within the plane of the primary bodies. We also assume that two of the primary bodies P2 and P3 possess equal masses (m2 = m3), and introduce a mass ratio parameter i =
"ia— [8].
m1+2m0
/
/
/
/
\ P \ •
\
o
Ps Z
Fig. 1. Dimensionless synodic coordinates for the restricted planar circular four-body problem. P1, P2 and P3 are the primary bodies, while P is the small-mass body
To describe the motion of the small body P, we introduce a right-handed synodic coordinate system Oxyz with its origin O located at the center of the line segment P2P3. The axis Oy passes through the body P1 and the axis Oz is perpendicular to the plane of motion. We then introduce dimensionless coordinates n (Fig. 1) with the coordinate transformation x = r£, y = rn, where r is the distance between the primaries. The equations of motion of the small body P take the following canonical form [10]:
dj _ dH dv dp^'
dn dv
dH dp/: dp ' dv
dH
'W
dpn dv
dH dn '
H = \ (pI + PV +pe U - ^(1 - 2ß) j -PvZ
JL
Pi
(2.1)
where
Pi = I C2 + n -
P2 H U +
1/2
+ n2
P3 H u -
1/2
+ n2
The independent variable v can be regarded as the angle of the Lagrange triangular rotation. In the problem considered we have v = wi, where w2 = f (m1+2m2)r-3 and f is the gravitational constant.
The system (2.1) has the following steady-state solution:
£ = 0, r] = ri„ pj: = + —(1 - 2fj,), pv = 0,
(2.2)
where n is a real root of the following equation:
r?-— (1-2^)-
4(1 - 2ß) sign (2n -(2r?-73)2
(4n2 + 1)3/2
The solution (2.2) describes a relative equilibrium located on the axis Oy of the coordinate system Oxyz. There exist three equilibria of the above type. They are denoted in Fig. 2 by points A, B, C. The equilibrium B exists for ¡i > 0.28379, whereas the equilibria A and C exist for any admissible values of ¡. In an equilibrium the small body P forms together with
2
2
1
1
2
2
"A
/
/
/
\
\
/
/
\
/
\
/
O
\
/
\
/
<'C
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
the primaries P\ (i = 1, 2, 3) a rigid central configuration which rotates in absolute space with constant angular velocity w.
For the relative equilibria considered and their respective central configurations to be stable, the primary bodies have to form a stable Lagrangian triangle. The stability of the Lagrangian triangular configuration is determined by Routh's necessary condition [11] that takes the following form:
For the central configuration PlP2BP3 inequality (2.3) is not fulfilled. Thus, it is always unstable. The stability analysis of the configurations PlP2 AP3 and PlP2CP3 has been performed in [8-10]. It was established that the configuration PlP2AP3 is unstable for any possible values of ¡. The configuration PlP2CP3 is unstable for i > ¡2 & 0.00270963 and it is linearly stable for 0 < i < ¡2. Thus, we are only interested in the central configuration PlP2CP3 and the relative equilibrium point C in what follows.
Lyapunov's theory shows [12] that in the small neighborhood of stable equilibria there exist periodic motions that form the so-called natural families classified by the periods of motions. These periodic solutions are known as Lyapunov motions. The motions with the smallest period TS are called short-periodic and exist for both stable and unstable equilibria. Depending on the problem's parameter values there can also exist one or more families of the so-called long-periodic motions with periods TL > TS. In a nonresonant case there exists exactly one family of long-periodic motions in the neighborhood of a stable equilibrium.
The objective of this paper is to obtain nonresonant families of periodic motions originating from the stable relative equilibrium C in the central configuration PlP2CP3 and to investigate their existence and orbital stability for all admissible values of the energy constant h and mass ratio parameter ¡ .
3. On the numerical continuation algorithm
To construct periodic motions outside of the small neighborhood of the equilibria, we use a numerical continuation algorithm which was first proposed by A. Deprit and J. Henrard in [14]
(2.3)
and developed further in [15, 16]. We assume that an autonomous Hamiltonian system
dH dH
Qi = -z—, Pi = , = h 2,
dPi dQi (3.1)
H = H(qi, pi, n),
with energy constant H = h and a parameter vector n = (nl, ..., nk)T admits a periodic solution
q* = q*(t,n*), p* = p*(t, n*), i = 1, 2, (3.2)
with period T* and initial conditions
q*o = q*o(0, п*), p*o = p*0(0, п*), i =1,2, (3.3)
corresponding to parameter values n*. The solution (3.2) with the initial conditions (3.3) is assumed to be known beforehand and will be called a base solution in what follows.
We introduce small finite parameter variations An, Ah and set out to find corresponding corrections £i0, ni0 (i = 1, 2) and t for the base initial conditions (3.3) and base period T* that yield initial conditions
qio = qio(0, n* + An), Pio = Pio(0, n* + An), i = 1, 2, (3.4)
and period T = T* + t of a new periodic motion
qi = qi(0,n* + An), pt = pt(0,n* + An), i = 1, 2, (3.5)
that corresponds to the parameter vector
n = n* + An (3.6)
and satisfies the so-called adherence conditions
lim qi(0, tt) = q*(0, n*), att^o
lim Pi(0, n) = p*(0, n*),
att^o
lim T = T .
att^o
To obtain the initial conditions (3.5) for the new periodic motions (3.4), we introduce local Cartesian variables
£i = qi- q*, ni = pi- p*, i =12, (3.7)
so the corrections £io, nio (i = 1, 2) take the following form:
£io = qio — qio, nio = pio — p*o, i = 1, 2. (3.8)
Applying the transformation (3.7) to the initial Hamiltonian equations (3.1) and linearizing the right-hand expressions, we arrive at a nonautonomous Hamiltonian system that describes motion in the neigborhood of an orbit described by the base solution (3.2). We then introduce a rotational coordinate transformation
Vi, = S
/ nu\
mu nv \mv/
(3.9)
that allows us to express the corrections (3.8) in terms of normal and tangential displacements w to the base periodic orbit. Following [15], we denote nu and nv as normal displacements, mu as a tangential displacement and mv as an isoenergetic tangential displacement. It has been shown in [16] that for a system with two degrees of freedom the symplectic orthogonal matrix S in (3.9) can be expressed in the following explicit form:
S = (R, s, -IR, -Is) =
/ P2 q* -q2 -P*\
P* q2 q* -P2
V
q2 Pî p2 q* V—qî P2 -Pî q2 )
(3.10)
where V = \/ql'2 + q22 + fil2 + fi22 and I is a symplectic orthogonal unit matrix. The asterisk symbol denotes that the marked values are calculated on the base solution. By applying the transformation (3.9) we obtain Hamiltonian equations for normal and tangential displacements:
dH" dn„,
dHn
Uv dn„, '
_ V
"u ~ V'
m„, = —mu + hunu + hM + ^ h^Airj + h^Ah,
j=*
(3.11)
with a first integral mu = y I Ah — j anc^ Hamiltonian
II" = g (/7nnl + /733«2 + ^nunvh13) +
k \ Ik
+ nu | £ hi*An- + hfr* Ah ) + nv I £ h\2Anj + hk+Ah j=* J \J=*
The time-dependant coefficients are given by the following formulae:
_ 1 ( , dH JPH d2H t d2H
'hi y [ -lh4^+P2g^- Plg^+q2g^r
1
3
dH
d2H
Qi-
2U' 3
d2 h
/7'12 V ( h'M chïj + & dq\dTT, ^1 dq^chr
I • *
+ P2
"j7 ( —h44
dH
P*
d2H
P2-
2W' j d2 h
I • *
+ q*
3
d2H
dpld-Kj
d2H dp\dn3
dn- dq*dn- dq2dn j = 1,...,k, hk+* = -h*4, hi+* = -h34, hk+* = h44,
q1
q2
d2H dp*2d ir^
d2H ' dp*2d ir^
d2H N
dpldiTj
h11 = RT(HR + IÈ), h13 = -RT(HIR - R), h14 = RTI(IHI + H)S, h33 = -RTI(HIR - R), h34 = -RT(IHIH) s, h44 = -f (IHI + H)s,
where H is the Hessian matrix of the Hamiltonian system (3.11). These time-dependant coefficients are calculated on the base solution.
1
To separate the coordinate displacements in accordance with the parameter variations, we represent the coordinates nu, nv, mu and the period deviation t as linear combinations of independent parameter variations An j (j = 1, ..., k) and energy constant variation Ah:
m,
it
nu = E nUAnj + nU+1Ah, J=1 k
mJuAnJ + mU+1 Ah,
i
u / J u J u
j=1
V = ^ nJv AnJ + nk+1Ah, J=1 k
T = ^ TJ AnJ + T k+1Ah.
J=1
(3.12)
Substituting (3.12) into (3.11) yields the so-called predictor equations [15, 16]:
( nii = hunu + h33n
11 11
633"-12j
n 12 = —h11 n11 — h13n12, n 21 = h11n21 + h33n22, „ n22 = —h11 n21 — h13n22, j = h13np1 + h33np2 + h12, j = 1, ...,k + 1, np2 = —h13np1 — h33np2 — h11, j = 1, . ..,k + 1
V
mi = y'mi + hunn + hMnl2,
V
m2 = y'm 2 + hUn21 + ^34w22>
™<2 = ^rn32 + h14n3pl + /';;1"/.2 + i = - +
nJu(0) =
nV(0) =
J
- n22{T)njpl{T) + n12(T)^2(T) nn(T)n22(T)-n12(T)n21(T)-nn(T)-n22(T) + 1;
rj^T) ~ nn(T)n3y2(T) + n2l(T)njpl(T) nn(T)n22(T)-n12(T)n21(T)-nn(T)-n22(T) + 1;
+ m2ni( 0) + m£(T)),
1
m„
V (0)
Ah-E
J=1
(3.13)
(3.14)
(3.15)
By solving Eqs. (3.13) and (3.14) with the initial conditions (3.3) and n11 = 1, n12 = 0, n21 = 0, n22 = 1, m1 = 0, m2 = 0, np1 = 0, np2 = 0, mPp = 1 and using (3.15), we first obtain the values (3.12) and then the corrections (3.8) that yield the approximate initial conditions (3.4) and an approximate period for the sought-for new periodic motion (3.5).
The motion given by the approximate initial conditions is characterized by the discrepancy value [18]
A = sj(qi(0) - qi{T)Y + (i2(0) - q2{T)Y + (Pl(0) -Pl(T)r + (p2(0) -p2(T)r, (3.16)
which describes the difference between the coordinate values over a period T, and by the energy constant discrepancy
Ah* = h - ha. (3.17)
The energy discrepancy (3.17) denotes the difference between the required value of the energy constant h and the actual value ha corresponding to the nonperiodic motion given by the approximate initial conditions.
To reduce these discrepancies, we set the parameter variations An and energy constant variation Ah to zero and look for the initial condition corrections (3.8), energy constant correction Ah* and period correction t that yield the improved approximate initial conditions (3.4). These corrections can be calculated using the so-called isoenergetic corrector [15, 16]:
npi = hanpi + + ^huAh*, np2 = -h13npl - h33np2 + ^h34Ah*, m2 = ym2 + hl4npl + yh44Ah*,
ni(0) =_n^m + (n22(T) - 1)N2_
j nn(T)n22(T) - nl2{T)n2l{T) - nn(T) - n22(T) + 1:' J
nn(T)n22(T) — nl2(T)n2l(T) — nn(T) — n22(T) + 1' j + (3.18)
= np2(T) - q*(0)Aq* + q** (0)Aq* + p*(0)Ap* + p*(0)Ap2, N2 = -npi(T) - p*(0)Aq* + p*(0)Aq* - q*(0)Ap* + q*(0)Ap2,
tanJ' = + m2 ni(0) + m|(T)), j = l,...,k + l,
Ah
mv = y^, Aq* = q*(T)-q*(0), Ap* = p*(T) - p*(0), ¿ = 1,2.
Solving (3.18) with the initial conditions (3.3) and n11 = 1, n12 = 0, n21 = 0, n22 = 1, m1 = 0, m2 = 0, njp1 = 0, np2 = 0, mp = 0 gives improved approximate initial conditions for the sought-for new periodic motion. Here (3.3) are approximate initial conditions obtained on a preceding predictor or corrector step. This procedure can be repeated until the discrepancy (3.16) becomes smaller than a small preset value 1 » e > 0.
The iterative computation procedure begins by setting the initial conditions (3.3) of the base solution (3.2), the global parameter variations An, AH and the parameter variation magnitude n0. The global parameter variations describe the destination point within the parameter space, while the variation magnitude is used to compute parameter variations An, Ah for the first iteration of the procedure. A single iteration of the computation procedure consists of two iterations of the predictor followed by one or more iterations of the corrector. The discrepancy values acquired on the two predictor steps are used to automatically estimate parameter variations An, Ah for the next iteration of the procedure. The computation terminates when it becomes impossible to obtain the solution for the normal displacement equations in the predictor. This occurrence is known as natural termination [19] and is usually associated with bifurcation or reaching an equilibrium. Following [15], the termination condition for the computation procedure can be expressed as follows:
| det(N(T) - E)\ < e,
where N(T) is the fundamental matrix of the system (3.13), E is the corresponding unit matrix and 1 » e > 0 is a small preset value defined by the required numerical precision. A more detailed description of the computation procedure is given in [16-18].
4. Linear orbital stability analysis
To draw conclusions on the linear orbital stability of periodic motions obtained by numerical continuation, we consider the differential equations for the normal displacements [15]:
n = IHnn, (4.1)
also given by (3.13). The characteristic equation of the system (4.1) is
p2 - 2Ap + 1 = 0, A = \(Sp[N(T)]) = \(nn(T) + n22(T)), (4.2)
where N is the fundamental matrix of the system (4.1) normalized by the condition N(0) = E and calculated for the T-periodic solution under investigation. If the condition < 1 is met, the solution is linearly orbitally stable. Since the system (4.1) is part of the predictor equations, the linear orbital stability analysis is carried out using the same computational procedure as the numerical continuation.
5. Analytical computation of nonresonant periodic motions
To obtain analytical expressions describing the Lyapunov periodic motions in the neighborhood of the relative equilibrium C under study, we use the linear normalization procedure as described in [13]. We first consider Eqs. (2.2) and introduce local canonical variables q1, q2, pi, p2:
£ = Qi, 1 = 1* + 92> P$ = -1* + —(1 - 2fj)+p1, pv = p2. (5.1)
We then expand the initial Hamiltonian (2.1) in a power series of the new canonical variables
H = H2 + H3 + H4 + ... (5.2)
where the quadratic part H2 takes the following form:
H'2 = \ (Pi +P2) + \ (aql + bq22) +j■)1q2- p2q1, (5.3)
with the coefficients a and b given by the formulae
8(—1 + 2fi) sign (2r? - y/3) 64fi (r?2 -a = : =73 I"
(-2r?+y3)3 (4r?2 + l)5/2'
16(-1 + 2^i)sign (277 — y/S) 64/x (—2rj2 + 7) b =---^ +
(5.4)
(-2r? + V3)3 (V + l)5/2
Since the expressions for higher-order terms H3 and H4 are cumbersome, we do not give them here.
If 0 < n < fi2, the characteristic equation of the linear system with the Hamiltonian H2 has two pairs of simple purely imaginary roots ±iwi and ±iw2, which implies that the equilibrium considered is Lyapunov stable. Given this fact and using the following linear change of variables:
qi = nnQi + nuQ2, Q2 = mnPi + muP2,
Pi = mjiPi + mj2 P2, P2 = njiQi + njj Q2, we bring the Hamiltonian of the problem to the normal form
H = IWl (Q'l + Pi) - \u2 (Q| + Pi) + H^, (5.5)
where
/1 , ,_x\l/2
i = i- [a, + b + 2 + (—l)i+1 \/(a, — 6)2 + 8(a + 6)J J , ¿ = 1,2, Xi K2 + b — 0 Xi - ab + a)
nli — " ! n2i —
mii = ¿iXi N2 - a - ^ , m2i = ¿iXi(a + b),
(
A —Onfh _ 1 v.,2
N
2
— I
y? — _^_ , I 2
At - (1 a ,1 9 , /7 ,1 / 9 , n . n\ ) — J-)
(5.6)
^ (bw| — 2a(b — 1)w2 + (b — 1) (a2 + 2a + b))
and H (3) are terms of order higher than two. To obtain the periodic motions, we consider the normal form (5.5) and arrive at the following expressions:
t c
q1 = —p(sin(i21z/) — cos^z/)),
* p1 = -^=(sin(Q1z/) +cos(Q1z/)), (5.7)
KP2 = 92 = 0,
c
q2 = —p(sin(f22l/) ~ cos(Q2z/)),
<
p2 = -^=(sin(Q2z/) + cos(Q2z/)), I Pi = qi = 0,
which describe families Fs and FL of short- and long-periodic motions with periods Ts = and TL = respectively. The frequencies Q1 and Q2 are given by the series
Q = + 0(c2), i = 1, 2, of the small amplitude parameter c = c(h).
6. Existence and orbital stability domains of the nonresonant periodic motions
Since the analytical representation of the periodic motions (5.7) and (5.8) obtained in the previous section only remains valid in a small neighborhood of the relative equilibrium C, we carry further computation and analysis using the numerical method described in Sections 3 and 4.
Fig. 3. Existence (grey) and instability (hatched) domains for the family FS of short-periodic motions emanating from the relative equilibrium C in the central configuration P1P2CP3
We first consider the short-periodic motions of the family FS and, using their analytical representation (5.7) as base solutions, we numerically compute their existence and linear orbital stability domains.
The existence and linear orbital stability domains for the family Fs of short-periodic motions emanating from stable relative equilibrium C associated with the central configuration P1P2CP3 are shown in Fig. 3. These motions exist between the curve C0 corresponding to the central configuration and the curve C3 where the family Fs terminates. The left-hand border is given by the condition i = 0, which corresponds to the restricted two-body problem. The periodic motions belonging to the family FS are linearly orbitally stable in the areas D1 and D2 and are orbitally unstable in the parametric resonance area D3 between the curves C1 and C2.
The left plot in Fig. 4 shows periodic orbits from the family FS for the fixed value of i = = 0.001 and three values of the energy constant h. The orbit S1 corresponds to the relatively small deviation Ah of the energy constant from its value for the central configuration with Ah & & 0.0001. The orbit S2 lies within the region D3 of the orbital instability and the orbit S3 is at the upper border C3 of the existence domain with Ah & 0.006. The right plot shows three orbits at the upper border C3 of the existence domain for the values of i = 0.0002 (orbit S4), I = 0.001 (orbit S3) and i = 0.002 (orbit S5). All orbits of the short-periodic family FS share the general shape shown in Fig. 4.
The long-periodic motions have also been considered, but only computed in a relatively small neighborhood of the considered relative equilibrium. The computation shows that in the neighborhood of the third-order resonance these periodic motions behave in accordance with the general analytical results obtained in [20-22] for Hamiltonian systems with two degrees of freedom. The subdomains D1 and D2 of the plot shown in Fig. 3 contain distinct families FL1 and FL2, respectively. The family FL2 terminates at the border C1 and the family FL1 terminates at the border C2 with the subdomain D3 containing both of these families.
To verify the results, Poincare maps have also been constructed for particular values of the problem's parameters. Figure 5 shows a Poincare map constructed for parameter values i =
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 the upper existence border for the values of j = 0.0002, j = 0.001 and j = 0.002, respectively
Fig. 5. Poincare map for p = 0.001 and h = —1.497 showing a stationary point PS corresponding to the orbitally stable periodic motion of the short-periodic family FS
= 0.001, h = —1.497 where PS is a stationary point corresponding to an orbitally stable periodic orbit of the short-periodic family Fs .
Conclusion
In this work we investigated the nonresonant families of periodic motions emanating from the relative equilibrium associated with a stable central configuration in the restricted planar four-body problem. The motions were obtained analytically in the small neighborhood of the
relative equilibrium and then continued to the borders of their existence domains using a numerical method. Existence and linear orbital stability domains of the short-periodic motions were constructed for all admissible values of the problem's parameters, while the long-periodic motions were investigated for particular values of the problem's parameters. The numerical results obtained herein are consistent with analytical conclusions obtained in earlier works.
Acknowledgments
The authors thank Prof. B. S. Bardin for discussion and guidance in preparing this work.
Conflict of interest
The authors declare that they have no conflict of interest.
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