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3Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 589-607. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221211
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 34D20, 37J40, 70K30, 70K45, 37N05
On the Orbital Stability of Pendulum Oscillations of a Dynamically Symmetric Satellite
B. S.Bardin, E. A. Chekina, A. M. Chekin
The orbital stability of planar pendulum-like oscillations of a satellite about its center of mass is investigated. The satellite is supposed to be a dynamically symmetrical rigid body whose center of mass moves in a circular orbit. Using the recently developed approach [1], local variables are introduced and equations of perturbed motion are obtained in a Hamiltonian form. On the basis of the method of normal forms and KAM theory, a nonlinear analysis is performed and rigorous conclusions on orbital stability are obtained for almost all parameter values. In particular, the so-called case of degeneracy, when it is necessary to take into account terms of order six in the expansion of the Hamiltonian function, is studied.
Keywords: rigid body, satellite, oscillations, orbital stability, Hamiltonian system, local coordinates, normal form
1. Introduction
A rigid body moving in a central Newtonian gravitational field can be considered as a good mathematical model for many problems of celestial mechanics and spacecraft dynamics. If the mass center of the rigid body moves in a circular orbit, then motions of the body about its center mass can be described by an autonomous Hamiltonian system [2]. As is well known, autonomous Hamiltonian systems admit natural families of periodic solutions, which are unstable in the sense of Lyapunov, because their period depends on initial conditions. In spite of Lyapunov instability,
Received November 10, 2022 Accepted December 04, 2022
This work was supported by the grant of the Russian Science Foundation (project No. 19-11-00116) at the Moscow Aviation Institute (National Research University).
Boris S. Bardin [email protected]
Evgeniya A. Chekina [email protected]
Aleksandr M. Chekin [email protected]
Moscow Aviation Institute Volokolamskoye sh. 4, Moscow, 125080 Russia
the above solutions can be orbitally stable, which is why they are of interest for applications in physics and mechanics. In spacecraft dynamics the orbitally stable motions can be regarded as working mode, in which a satellite is stabilized by means of gravity forces. In particular, orbitally stable modes can be found in the class of planar motions, when one of the principal axes of inertia of the satellite is orthogonal to the plane of the circular orbit.
The study of the stability of planar periodic satellite motions has been started in [3-5], where the stability analysis was performed using a linear approximation. The limiting case of this problem, when the period of planar motions is much greater than the period of revolution of the center of mass in the orbit, was investigated in [6]. Comprehensive studies of the linear stability of planar periodic satellite motions for some particular cases of satellite mass geometry have been performed in [7, 8].
A nonlinear analysis of the problem of the orbital stability of planar periodic motions was first carried out by A. P. Markeev in [9], where an oblate dynamically symmetric satellite was considered. In particular, it was proposed to transform to action-angle variables on the invariant manifold corresponding to the planar periodic motions. This allows one to introduce proper local variables in the neighborhood of the unperturbed periodic orbit and to write down the equations of perturbed motion as a Hamiltonian system with two degrees of freedom. The stability of this system has been investigated on the basis of KAM theory by calculating the normal form of its Hamiltonian. This approach was applied for a rigorous study of the orbital stability of planar periodic motions both for dynamically symmetric satellites [10, 11] and for satellites with unequal moments of inertia [12]. Later on, a comprehensive nonlinear analysis of the orbital stability of planar rotations and oscillations of a plate-shaped satellite was performed in [13-16]. In those papers, a nonlinear stability analysis was performed for the general position case, when it is enough to calculate the Hamiltonian normal form through terms of the fourth order, to apply KAM theory and obtain rigorous conclusions on stability. Special cases of degeneracy, when it is necessary to normalize the Hamiltonian through terms of order six or higher, have not been considered due to some technical difficulties in calculating explicit expressions for the coefficients at higher-order terms of series expansion of the Hamiltonian of the perturbed system.
In [1], a new method is proposed to introduce suitable variables in the neighborhood of periodic orbits of a Hamiltonian system. In accordance with this method, such variables can be introduced by means of a nonlinear canonical transformation, which is constructed in the form of power series in a new variable (the so-called local variable). This provides a way around the above-mentioned technical difficulties and allows one to obtain an explicit form of series expansion of a Hamiltonian of perturbed motion through terms of any finite order. The above method has been applied in the problem of the orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev-Steklov case [1, 17].
In this paper, using the approach developed in [1], we study the problem of the orbital stability of planar oscillations of a dynamically symmetric satellite. In particular, we focus on degenerate cases when the terms of order six should be taken into account to answer the stability question.
2. Problem statement
Let us consider the motion of a dynamically symmetric satellite about its center of mass O, moving in a circular orbit in a uniform field of gravity. Let OXYZ be an orbital coordinate system whose axes OX, OY, OZ are directed along the radius vector of the center of mass O,
the transversal and the normal of the orbit, respectively. A relative coordinate system Oxyz is rigidly attached to the satellite and is formed by its principal axes of inertia (the axis Oz is directed along the symmetry axis of the inertia elipsoid); A = B and C are the corresponding moments of inertia. We will specify the orientation of the satellite in the orbital coordinate system OXYZ using the Euler angles 0, p and introduce the corresponding generalized momenta p^, pe, p^. The equations of body motion can be written in canonical Hamiltonian form with the following Hamiltonian function [10]:
n = o a + \ fcot'2 9 + Pi + 7T ~ -^TZV^Vv ~ P-<p + ?(<* - 1) sin2 ip sin2 6, (2.1) 2 sin2 0 2 \ a J ^ 2 sin2 0 v ^ v 2
where a = ^ and the true anomaly v is taken as an independent variable.
The equations with Hamiltonian (2.1) admit a family of particular solutions on which
n
0=2> ^ = 0> Pf = Pe = 0> (2-2)
and the evolution of the variables p^ is described by canonical equations with the Hamiltonian
1 3
= 2*4 - H + 2" 1} sin2 ^ (2-3)
These solutions describe planar motions of the satellite in which its axis of symmetry lies in the orbital plane of its center of mass.
Let us note that the angle p is a cyclic coordinate. Hence, p^ is a first integral of the canonical system with Hamiltonian (2.1), that is, pv = const in the motion of the satellite.
In what follows, we will assume that the relation pv = 0 is satisfied for the full system with Hamiltonian (2.1), that is, we consider the problem of stability on the level set of the first integral pv, where the planar motion lies.
For ease of further calculations, we consider two cases separately: the case of a prolate satellite (0 < a < 1) and the case of an oblate satellite (1 < a ^ 2). Let us introduce new coordinates q1, q2 and momenta p1, p2 for the case 0 < a < 1 by the following formulae:
and for the case 1 < a ^ 2 by the following formulae:
91 = q'2 = e~^ = Y P2 = 7^ry (2"5)
and transform to a new independent variable r = \/3\a — l\v.
Next, we present calculations necessary to study the orbital stability of planar oscillations for the case of an oblate satellite (1 < a < 2), which has turned out to be more interesting. In the case of a prolate satellite, all calculations are carried out similarly. In variables qi, pi (i = 1, 2) the motion of the satellite is described by the canonical equations
dqi dH dpi dH
it-BVt- it - Bq, << = 1'2> (2-6)
with the Hamiltonian
pf tan2 q2 f p, 1 \ pi 1 ,2 2
H =-^----. —pz -\--, M—£ + - sin qi cos öo. (2.7)
2cos2 q2 s/^T\ s/3 6y/<x=Tj 2 2 v '
The Hamiltonian (2.7) was obtained by substituting (2.5) into (2.1). _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(4), 589 607
For the above-mentioned planar motions we have
Qi = f (r), Pi = ), Q2 = P2 = 0, (2.8)
where the functions f (t), g(r) are a solution of the canonical system with the Hamiltonian
H^ = \p\ + \ sin2 (2.9)
Depending on the constant of the energy integral H (°) = h, the above solution describes either planar periodic motions or a motion in which the satellite asymptotically approaches its unstable equilibrium position in the orbital coordinate system. If the constant of the energy integral is h0 < then the solution of the system with the Hamiltonian (2.9) describes planar pendulum oscillations with amplitude tp0 and frequency Q, = 2K{k)' ^ = sin 0o- is a complete elliptic
integral of the first kind.
In this case the functions f (t), g(T) read
f(t) = arcsin[ksn(T, k)], Jy ' L V n (2.10) g(T) = k cn(T, k).
The problem under consideration has two parameters: the inertial parameter a and the amplitude of oscillations , which is a parameter of the family of periodic solutions (2.10).
The purpose of this work is to carry out a rigorous nonlinear analysis of the orbital stability of the above-mentioned planar oscillations. To solve this problem, we apply the method of introducing local variables developed in [1]. This approach allows one to study the previously unconsidered cases of degeneracy when the terms of order six in the expansion of the Hamiltonian in a neighborhood of the periodic orbit must be taken into account to answer the question of orbital stability. We also correct some calculation inaccuracies of [10].
3. Equations of the perturbed motion. Isoenergetic reduction
In order to study the motion in a neighborhood of the unperturbed periodic orbit, we perform the following canonical change of variables q1, p1, q2, p2 ^ n, Q2, P2:
qi = f (t ) + )n + a2(( )n2 + as (On3 + .. • Pi = g(t) + MOn + )n2 + bs(t )n3 +...
(3.1)
The algorithm for constructing power series (3.1) up to any finite degree of n has been proposed in [1]. The coefficients ai(t), bi(t) (i = 1, 2, 3) calculated by means of this algorithm read
ai(t) = -
J_dg
v^ie 1 df
a2(t) =
b2(t) =
2V4 dÇ2 ' 1 d'2g 2V^dë'
as (t) =
1 d3g W2 dg
bs(t) = -
6V6 dt3 1
d3f W2 df
6 V6 dt3 3 V8 dt '
(3.2)
where
V2 =
+
dg
Ue
+
22
de
(3.3)
2
2
2
In the new variables the family of periodic solutions (2.8) has the form
Z = T + Z (0), n = Q2 = P2 = 0. (3.4)
It is clear from (3.4) that the behavior of the trajectories in the vicinity of the considered periodic orbit can be well described by the variables Z, n, Q2, P2, which we call local variables. The problem of the orbital stability of the periodic solution (3.4) is equivalent to the problem of Lyapunov stability with respect to the variables n, q2, P2.
Let us now recall that the functions f (£), g(Z) are a solution (see (2.8)) of the canonical system with the Hamiltonian H. That is why the following relations take place:
£=* 0 = (8.5)
d2 f 1 d2a
d3 f d3g
il cos 2/, = 2g2 sin 2/ + - sin 4/. (3.7)
Taking Eqs. (3.5)-(3.7) into account, we obtain the following explicit form of the canonical change of variables (3.1):
a -f + ^n2 + C^2 (cos 2/ + 4g2) + 2W2) sin 2/ 3
+ 2F2 4F4 1 + 12F8
/; „ . JL„ ££^„2 + ^(^2cos2/ + 2^2)
Pi -fif+ 2F4 1 6F8 1 + ),
where V2 = g2 + \ sin2 2/ and W2 = g2 cos2 2f + \ sin2 2/.
It is worth recalling that f and g are periodic functions of their argument. Thus, the coefficients of series (3.8) are periodic in the new canonical variable £.
Now we make one more univalent canonical change of variables Z, n — w, r by the formulae
£ = ^w, r] = fir. (3.9)
By substituting (3.8) and (3.9) into (2.7), we obtain the Hamiltonian of the perturbed motion in the neighborhood of the periodic orbit
r = r +r4 + r6 + ... + +..., (3.10)
where T2m is a form of degree 2m in q2, p2, r1/2. The forms r2, r4 and r6 of the expansion (3.10), which are required for further analysis, read
r2 = fir + ^2° (w)q2 + ^02 (w)p2, (3.11)
r4 = X(w)r2 + ^2° (w)Q2r + ^40 (w)Q4, (3.12)
r = ^(w)r3 + ^2°(w)q2r2 + ^4°(w)q4 r + ^6° (w)q6 (3.13)
The coefficients , ^, 0-, % and a are 2^-periodically dependent on w. They are calculated by the formulae
Q
X(w) = -^T74 (cos2/* " !) " V'2)
2V 4 Q3
a(w) = (2U4 + V2 cos 2/, - 6V2g2 + 4gf - W2)
Q
9*
= ^ + ^^
, , , Q / 2 2g,, '040 M = ^2 I V* +
= g
9* +
v/3 a — 3 1
\/3a; — 3
- V2
+ V2
= l{g*+ vra
- sin2 f
1
. ¥>02 (w) = 2'
--(cos2/,-l),
Mw) = -
cos 2f* - 1 + 17( 9* + Q2
\/3a; — 3,
(?)V2 \/(a — l)(cos 2/* - 1) + 4, 73 cos 2/,),
(3.14)
where /*, g^ denote the functions f*(w) = / (Q w), g^(w) = g(Q w), 2^-periodic in w.
By virtue of the equations of motion with Hamiltonian (3.10), the coordinate w is an increasing function of the variable t. Therefore, in the problem of the stability, to describe the motion on the zero isoenergetic level, we take the coordinate w to be a new independent variable. In addition, from the equation r = 0 with small q2, p2, r we have r = —K(q2, p2, w). The function K(q2, p2, w) is the series
K = K2 + K4 + KR + ... + K2m +
(3.15)
where K2m is a form of degree 2m in q2, p2 with coefficients 2^-periodic in w. The forms K2, K4 and K6 have the following explicit form:
K2 = ^2 (Q2, P2, w),
K4 = ^4092 - ^20K292 + XK2 >
K6 = (^60- ^4o02o)q| + K2 (020- ^40) q4-- K22(X020 - «20)92 + K2 (2XK4 - K .
(3.16)
(3.17)
(3.18)
The equations of motion on the isoenergetic level r = 0 can be written in the Hamiltonian
form
dq2 dK dp2
dw dp2
dw
dK
dq2'
(3.19)
Thus, the problem of the stability of pendulum-like periodic motions of the satellite reduces to a stability analysis of the equilibrium point q2 = p2 = 0 of the reduced system (3.19).
2
2
2
1
4. Linear orbital stability
Important conclusions on the stability of the equilibrium q2 studying the following linear canonical system:
p2 = 0 can be obtained by
dK2
dq2 =_
dw dp2
dp2 dw
dK2 dq-2
with Hamiltonian (3.16). The characteristic equation of system (4.1) reads
p2 - 2Ap + 1 = 0,
(4.1)
(4.2)
where 2A = xn(2n) + x22(2n) is the trace of the matrix X(w), which is the matrix of the fundamental solutions of system (4.1). It satisfies the initial conditions X(0) = E2, where E2 is the second-order identity matrix.
If |A| > 1, then the equilibrium q2 = p2 = 0 is unstable both in the linear system (4.1) and in the nonlinear system (3.19) [18]. Thus, in this case, the corresponding periodic oscillations are orbitally unstable. Otherwise, if |A| < 1, then the equilibrium q2 = p2 = 0 is stable in the linear system (4.1), but this does not mean its stability in the full nonlinear system (3.19).
1.6-1
0 0.2 0.4 0.6 0.8
a
Fig. 1. Linear stability diagram for pendulum-like oscillations at 0 < a < 1
In the general case, the values of the coefficient A can be obtained only numerically. The linear stability analysis based on numerical calculations of A was provided in [10]. Here we briefly review the results in [10], which are represented in Fig. 1 and Fig. 2. Gray denotes the region of orbital instability, where |A| > 1. It represents a countable set of instability zones emanating from the points a^ = 1 — |(??2 + 4??,), n £ N and a^ = 1 + + l)2 of the a axis. These
exact values for a« and ai2^ were obtained analytically regarding the amplitude as a small parameter of the problem. The instability zones become narrower and, as n tends to infinity,
they approach two straight lines a = 1, tp0 = . At 1 < a < 2 most of the boundary curves of
1.61
V'o 0.8
0.4-
0.6-
0.2-
1.4-
1.2-
0
1-
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
a
Fig. 2. Linear stability diagram for pendulum-like oscillations at 1 < a < 2
the instability zones go to the point The first four curves shown in Fig. 2 from right to
left are the only exception from this rule: two of them go to the point (a1, and two others go to the point (a2, §), where a1 & 1.34276 and a2 ~ 1.60413. Outside the instability zones, at | A| < 1, the linear system is stable, that is, the planar oscillations are linearly orbitally stable. However, the stability in the linear system does not guarantee the orbital stability in the original nonlinear system. That is why it is necessary to carry out a nonlinear analysis in order to obtain rigorous conclusions on orbital stability outside the instability zones.
5. Orbital stability study in the case of small amplitudes of the satellite oscillations
In this section, we perform a nonlinear stability analysis under the assumption that the amplitude of the satellite oscillations is small enough, that is, ^ 1. In this case, we apply the method of a small parameter to perform an analytical study of the orbital stability.
In accordance with the general approach for the nonlinear stability study of a Hamiltonian system, we have to construct a canonical change of variables which reduces the Hamiltonian (3.15) to the simplest form, the so-called normal form. The problems of stability for original and normalized Hamiltonian systems are equivalent. On the other hand, as follows from KAM theory, the condition of stability (or instability) can be represented in the form of inequality, which is written in terms of the coefficients of the Hamiltonian normal form. Usually, it is enough to calculate the normal form through terms of the fourth order. However, there are some special cases of degeneracy when it is necessary to normalize the Hamiltonian up to some order higher than four. Below we show that in our problem, for some parameter values, it is necessary to have the normal form of the Hamiltonian (3.15) calculated up to terms of order six to solve the problem of orbital stability.
It was shown in [10] that, at 0 < a < 1 in the case of small amplitude, the satellite oscillations which are linearly stable are also nonlinearly stable. In what follows we study the case 1 < a < 2.
Using the well-known expansions of elliptic functions [19], we can obtain expansions of the terms K2, K4 and K6 in the form of convergent series with respect to the amplitude which is regarded as a small parameter. In particular, the expansion for K2 reads
K2 = ¿A0 (pf + Ql) + 00 cos wQl+
+ [(12(a - 1) cos 2w + 1 )Q% + Pf] + -¿^(cosw + 3 cos 3w)Q%-
8 48
-¿0oAo [(24(a — l)(2cos 2w — cos4w + 3) — 11)Q2 — 11-Ff] + O (0o)> (5.1)
where A0 = (3a — 3)_1/2. Q2 and P2 in (5.1) denote new canonical variables introduced by the following formulae:
<72 = 00^2, = (5.2)
A0
The transformation (5.2) changes the scale of the canonical variables and normalizes the autonomous part of K2.
Now we continue the normalization of (5.1) with the aim to reduce it to the form corresponding to a harmonic oscillator.
First we transform to polar canonical variables 0, r using the formulae
Q2 = \/2rsin(/>, P2 = \/2rcos</>. (5.3)
Now the Hamiltonian reads
K = G2(0, w, ^o)r + G4(4>, w, ^0)r2 + G6(0, w, ^o)r3 + ... (5.4)
The function G2 has the following form:
G2 = A0 + 2sin2 0cos w^0+
+ 3A0 ((a — 1) cos 2w sin2 0 + t- ) 0o + 7 sin2 <fi cos w (3cos2 w — 2) + \ 12/6
+ T7S7 (48(« - 1) (12 cos4 w - 14cos2 w + 1) sin2 </> + 11) + O (5.5)
192
The functions G4 and G6 are forms in sin 0, cos 0 of order four and six, respectively. The coefficients of these forms are 2^-periodic in w and analytic in . The formulae for these coefficients are given in Appendix A.
By a proper choice of new canonical variables r, 0 ^ p, 0, we can exclude the dependence on w in the second-order terms of the Hamiltonian (5.1) and reduce it to the following normal form:
K = \p + Fa(0, w)p2 + F6(0, w)p3 + O (p4), (5.6)
where
A0(15a — 19) 2 . A-Ao+ 2(6«-14) 00 +
An (10 287a4 - 37881a3 + 52 677a2 - 36 939a + 12 560) 4 , , N
+ ~-192(3a-4)(3a-r)»-^+ <>«). (6.7)
The new variables 9, p can be introduced by the following generating function:
5*2 = p$($, w, ),
where $ is analytic in that is,
W, ) = $ + W) + %2$2($, W) + ...
(5.8)
(5.9)
In Appendix B, an algorithm is described which gives explicit expressions of $n($, w) up to any finite order.
The generating function (5.8) defines the following relations between old and new canonical variables:
d $
9 = $(</>, w, 0O), r = Pt^(</>, w, 0o)- (5-10)
Finally, from (5.10) the change of variables r — 9, p can be written in the following form:
$ = 0(0,w,$o), p = r
w, 0p) dcf)
-i
(5.11)
¿=©(9,W,%)
The function 0 is inverse to the function In particular, its series expansion in $o reads
6(0, w, $o) = 0 - $,$i(0, w) - $o2
* ta ^ d^l(0, w)^fû '
$2(0, W)--aF^ l( '
+
(5.12)
By taking into account (5.12) we have the following formulae for coefficients in the expansion of the Hamiltonian (5.6):
Fn(0, w, $o) = Gn(Q(0, w, $o), w, $o)
w, ip0) dcf)
(5.13)
¿=©(e,w,%)
In what follows we consider the nonresonant case, that is, the case where the inequality m\ = 2N (m = 1, 2, 3, 4; N £ N) is satisfied. In this case, by the canonical change of variables p, 9 — R, $ the Hamiltonian can be brought to the following normal form:
K = XR + c2R2 + F6(&, w)R3
where
2n 2n
c2
4n2
J J F4 (0, w) dwd0. oo
(5.14)
(5.15)
The change of variables p, 9 — R, which reduces the Hamiltonian (5.6) to the form (5.14), is a near-identity transformation analytic in and p and 2^-periodic in w and This transformation can be defined by the generating function
S4 = 0R + R2U(0, w). The function U(0, w) is calculated by the following formulae:
U(0, w) = Y(0 - Xw, w) + v(0 - Xw),
(5.16)
(5.17)
n
1
where
w
Y(90,w) = j(c2 — F4(Xu + d0,u)) du (5.18)
0
and the function v(9 — Xw) is chosen to satisfy the condition of 2^-periodicity of the function U (9, w) in w and 9.
The function F6 in the Hamiltonian (5.14) is calculated by the formula
dU
F6(§, w) = F6(§, w) + 2F4(§, w) — (§, w). (5.19)
Calculations show that by taking into account in F4(9, w) terms through order ^ we have
„ (9a2+ 9«-22) A03 2 2 4(3a — 7)
(972a5 + 31347a4 - 158 301a3 + 285 489a2 - 230 211a + 71152) An3 4
32(3a - 4)(3a - 7)3
+ О{ф60). (5.20)
If c2 = 0, then, by the Arnold-Moser theorem [20, 21], the equilibrium point of system (3.19) is Lyapunov stable. This implies that the oscillations are orbitally stable. But if c2 = 0, then the so-called case of degeneracy takes place and solving the problem of stability requires an additional analysis including terms of degree R3 in the Hamiltonian (5.14).
By solving the equation c2 = 0 with respect to a we have
1 V97 (655 6463\/97\ (2
a* =---Ь -------— \ фп + О Ш). (5.21)
2 6 V 144 13 968 Г v '
Now we can consider the case of degeneracy which takes place at a = aM. To this end we normalize the Hamiltonian through terms of order six. We can do this by analogy with the normalization of fourth-order terms. In particular, we perform a canonical near-identity change of variables ft, R ft, R which is analytic in and R and 2^-periodic in w and ft.
In the new variables the Hamiltonian reads
K = AR + c3R3 + O[R4 ), (5.22)
where
2n 2n
1
4n2
nn
c3 = —J / / F6(§, w)dwdd. (5.23)
Calculations show that
Since the quantity c3 is nonzero for sufficiently small , it follows that, by the Arnold-Moser theorem, the equilibrium point of system (3.19) is Lyapunov stable, and hence the corresponding pendulum-like oscillations are orbitally stable.
Now we note that in resonant cases a special study is necessary to solve the problem of orbital stability. In fact, it is enough to investigate only cases of third- and fourth-order resonances. The investigation performed in [10] has shown that at these resonances the satellite's oscillations with small amplitudes are orbitally stable. Thus, combining the results of this section with the results of [10], we can formulate the following conclusion on stability. For sufficiently small amplitudes, pendulum-like oscillations of the symmetric satellite which are linearly orbitally stable are also nonlinearly orbitally stable.
6. The nonlinear study in the case of arbitrary amplitude of satellite oscillations
In this section we study the orbital stability of pendulum-like oscillations for arbitrary values of the amplitude. We perform this study using the approach developed in [22]. The main idea of this approach is to construct a symplectic map generated by the system of nonlinear equations (3.19) and to explore the stability of its fixed point. The problem of stability of the fixed point of this map is equivalent to the problem of stability of the equilibrium position of system (3.19).
At first, we perform the following linear change of variables:
Q2 = nuQ + n 12 P, P2 = «21Q + n22P,
(6.1)
where
«11 = ^12(2tt), n12 = 0, n21 = A - xn{2ir), n22 = \/l- A2, A = ±[xn(2ir)+x22(2ir)].
(6.2)
The functions xn(w), x12(w), x21 (w), x22(w) are elements of the fundamental matrix X(w) of the linear system with Hamiltonian (3.15). In the new variables the Hamiltonian reads
K*(Q, P, w) = K^(Q, P, w) + K%(Q, P, w) + K*6(Q, P, w) +
(6.3)
where K2m(Q, P, w) is a form of degree 2m, Hamiltonian (6.3) is obtained by substituting (6.1) into (3.15) multiplied by (n11 n22)_1. In the variables Q, P the linear part of the symplectic map generated by the canonical system with Hamiltonian K*(Q, P, w) has the simplest form. The symplectic map can be constructed in the following way [22].
Let Q0, P0 be the initial values of the variables Q, P, and Q1, P1 be their values at w = 2^. Then the symplectic map generated by the canonical system with Hamiltonian K*(Q, P, w) reads
Q1 = G
P1
n I ^F, dF,
vo dp0 "t" dp0 dq0 dp0
dF, dPn
+ O6
io
u
p I dF4 iPF.dF, dF, n
0 + 9q0 -dqj 9p0 9q0 + u>
6
(6.4)
where O6 denotes terms of degree six or higher, Fm = $m(Q0, P0, 2^), and $fc(Q0, P0, w) are forms of degree m (m = 4, 6) satisfying the equalities
dw
— —G4,
_ 0G4
— —Cj6 —
dw
dP0 dQ,
(6.5)
0
Gm(Q0, P0, w) are forms which are obtained from Km(Q, P, w) by the change of variables
= X*(w)
Qo P.
(6.6)
where X*(w) is the fundamental matrix of the linear system with the Hamiltonian K| (Q, P). The matrix G reads
G
cos 2nß sin2nß — sin2nß cos2nß
ß = -7- arccos A. 2n
(6.7)
Equating coefficients of the same powers on both sides of Eq. (6.5), we obtain five ordinary differential equations for the coefficients of the forms (m = 4, 6). The right-hand sides of these equations depend on x^(w), which are entries of the matrix X*(w). Thus, integrating the system of sixteen equations (twelve equations for coefficients of the forms and four equations for Xj(w)) in the interval [0; 2n], we obtain the coefficients of the form Fm. In the general case, the above system should be solved numerically.
Let us introduce the following notation:
= 3/40 + /22 + 3/04, v4 = /22 _ /40 _ /04, ^4 = /13 _ /31,
1
5
1
¿6 = -3/06 - 2/24 - 3/60 - 2^2 + /3i(4/o4 + 2/22 + 5/40) + /i3(4/40 + 2/22 + 5/04)- (6.8) [(/04 " /22 + /40)2 + (/13 " /3l)2] cot 27Tß - [(/13 + /31)2 + 4(/04 - /40)2] cot 7T/3,
where /^ are the coefficients of the forms Fm
(m = 4, 6).
Now we briefly formulate stability and instability conditions for the fixed point of the map (6.4) obtained in [22-24]. Assume that there are no resonances up to fourth order in system (3.19), that is, the roots of the characteristic equation (4.2) satisfy the relation pk = 1 (k = 1, 2, 3, 4), where k is the order of resonance. In this case the inequality 54 = 0 guarantees the stability of the fixed point of the map (6.4). Otherwise, if 54 = 0, then the case of degeneracy takes place and an additional nonlinear analysis must be performed to draw conclusions on stability. In particular, it is necessary to calculate S6. If S6 = 0 and in system (3.19) there are no resonances up to sixth order, then the fixed point of the map (6.4) is stable [24, 25]. Otherwise, if S6 = 0, then terms of order eight or higher in the Hamiltonian expansion (3.15) must be taken into account to answer the question of stability. The cases of first- and second-order resonances take place on the boundaries of stability domains. The problem of the orbital stability of satellite oscillations at first- and second-order resonances has been studied in [11]. In the problem under consideration, the map (6.4) does not include terms of second order, which is why the condition of stability for the fixed point in the presence of third-order resonance is the same as in the nonresonant case, that is, S4 = 0. The case of fourth-order resonance should be considered separately. It takes place if the coefficient A of the characteristic equation (4.2) is equal to zero. In this case the fixed point of the map (6.4) is stable if the inequality |£4| > sjz/f + ¡x\ is satisfied. If |£4| < ^z/f + ¿i2, then the fixed point is unstable. In the case of degeneracy, when |1 = \/z/| + an additional nonlinear analysis is necessary to obtain conclusions on stability.
For the parameter values from the linear stability zones shown in Fig. 1 and Fig. 2 the calculation of the coefficients of the map (6.4) was performed numerically. It is worth noting
Fig. 3. Nonlinear stability diagram for pendulum-like oscillations 1 < a < 2
that the right-hand sides of system (3.19) have singularities at a = 1 and at ip0 = To avoid difficulties of numerical integration, we have omitted the calculation for the parameter values close to a = 1 and tp0 = |. In particular, we have restricted our study to the five widest stability zones
shown in Fig. 1 and Fig. 2 and calculated the coefficients of the map (6.4) for tp0 e [0; | — 0.01]. Using numerical calculations of the coefficients, we have checked the above-mentioned conditions for stability and instability and obtained the following conclusions on stability. At 0 < a < 1 the orbital stability of planar oscillations of the satellite takes place for all parameter values from the linear stability zones shown in Fig. 1. This conclusion refines the result obtained in [10]. Figure 3 shows a diagram of nonlinear stability study for 1 < a < 2. By y1 and y2 we denote the curves where the case of degeneracy takes place. These curves intersect the resonant curves at the points Ri (i = 1, ..., 5). The curves corresponding to resonances of the fourth order are shown as dashed lines and the curves corresponding to resonances of the sixth order are shown as dash-dotted lines. It appears that the planar oscillations of the satellite can be unstable in the presence of the fourth-order resonance. In particular, orbital instability takes place in the segment of the resonant curve bounded by points B1 and B2. Outside this segment and points Ri (i = 1, ..., 4) the planar oscillations are stable. At points B1, B2, Pj (j = 1, ..., 3), Ri (i =
= 1, ..., 4) corresponding to degeneration in the presence of resonance an additional stability study is necessary. The parameter values corresponding to these points are given in Table 1.
Table 1. The parameter values corresponding to the special points
Pi P2 PS Ri i?. Rs i?4 Rb Bl B2
a 1.2680 1.1487 1.1861 1.1492 1.1635 1.2235 1.2582 1.2446 1.2439 1.2456
V'o 1.2621 0.1186 1.2014 0.870 1.0552 1.046 1.2194 1.1559 1.1502 1.1648
Conclusions
The behavior of trajectories in the neighborhood of periodic motion of a Hamiltonian system can be investigated in detail by a proper choice of local canonical variables. In the study presented above the local variables have been introduced by a nonlinear canonical change of variables given by power series. Due to such an approach the equations of motion of a dynamically symmetrical satellite have been written in the form of a canonical system whose Hamiltonian function is a power series in the local variable. The coefficients of this series have been calculated in an explicit form using a constructive algorithm developed in [1].
On the basis of the above equations of motion, a comprehensive investigation of the orbital stability of the planar pendulum-like satellite oscillations has been performed. The orbital stability analysis given in this paper refines the results of [10]. In particular, some imperfections in numerical calculations of the above paper have been corrected and a set of parameters corresponding to orbital instability has been found. This set is represented in the stability diagram as a segment of the fourth-order resonance curve. Moreover, the approach used in the present study allowed us to solve the problem of the orbital stability of pendulum-like periodic motions in degeneracy cases, when it is necessary to perform a nonlinear analysis taking into account terms up to the sixth order in the expansion of the Hamiltonian.
Appendix A. Formulae for terms of the fourth an sixth order in the expansion of the Hamiltonian in polar coordinates
The functions G4(0, w, and G6(0, w, in the Hamiltonian expansion (5.4) have the following form:
where
G4(0, w, 00) = ^40(00, w) sin4 0 + g22("00, w) sin2 0cos2 0 + g04("00, w) cos4 0,
940(w, 00) = A0g40)(w)00 + A0g40)(w)0o + A0g430)(w)00 + A5g4J(w)04 + O 0),
g40) (w) = —2cos w, g40) (w) = 2 (12 cos2 w — 5) (a — 1) + sin2 w (2 cos2 w — 1),
940 (w) = —^ (l8cos2w — 13) (a — 1) + 60cos4 w — 93 cos2 w + 35) cosw,
g4o (w) = —6 (12 cos4 w — 13 cos2 w + 2) (a — 1)2 — —6(12 cos6 w — 20 cos4 w — 9 cos2 w + 12) (a — 1) —
37 33
—4 cos8 w + 11 cos6 w —— cos4 w — 41 cos2 w + —;
g22(w, 00) = A0g22)(w)00 + A0g22)(w)0o2 + A0g2|)(w"3 + A3g22)(w)04 + O 05),
g22) (w) = —2cos w, g22) (w) = —2 (2 cos2 w — 1) (cos2 w + 3a — 4),
922 (W) = (60cos4w — 93 cos2 w + 35) cos w,
g22 (w) = sin2 w (2cos2w — 1) (6 (3 cos2 w — 1) (a — 1) + 4 cos4 w — 5 cos2 w + 2) ;
where
gQA(w, ) = AOgiJ+ AO+ O ($), ffof (w) = sin2 w (2 cos2 w — 1) ,
9(H (w) = ^ sin2 w (2 cos2 w — l) (4 cos4 w — 5 cos2 w + 2);
G6(0, w,^0) = g60(^0, w)sin6 0 + g42(0, w)sin4 0 cos2 +g24(0, w) sin2 0 cos4 0 + g06 (00, w) cos6 0,
960(w, = A0g60)(w)^0 + A0g60) (w)^3 + A5g6o)(w)^o + O (^5), g60) (w) = — cos w,
#60 (w) = ^ (8 (24 cos2 W - 17) (a - 1) + 40 cos4 w - 65 cos2 w + 23) cos w,
g(6o{w) = ^ (540 cos4 w - 555 cos2 w + 77) (a - 1)2+ 5
+ (216 cos6 w — 400 cos4 w + 201 cos2 w — 20) (a — 1)+
+ - sin4 w (6 cos4 w — 6 cos2 w + l); 3
g42(w, = A0g42)(w)^0 + A0g42)(w)$3 + A0g42(w)^4 + O (i>0),
g42) (w) = —2cos w,
942 (w) = ^ (^ (18 cos2 w — 13) (a — 1) + 40 cos4 w — 65 cos2 w + 23) cos w,
g42) (w) = 12 (12 cos4 w — 13 cos2 w + 2) (a — 1)2+
+ (288 cos6 w — 544 cos4 w + 285 cos2 w — 32) (a — 1) — — 4 sin4 w (6 cos4 w — 6 cos2 w + 1);
g24(w, = A0g24)(w)^0 + A0g24) (w)$3 + A0g24(w)^04 + O ($),
g24) (w) = — cos w,
9m (w) = ^ (40 cos4 w — 65 cos2 w + 23) cos w,
g24) (w) = —4 sin2 w (6 cos4 w — 6 cos2 w + 1) (cos2 w + 3a — 4);
g06 (w ^0) = A0g0e) (w)^0 + O ,
(w) = ^ Sin4 W (6 cos4 W — 6 cos2 w + l) . 3
Appendix B. The algorithm for calculating the generating function S2
Let us consider a canonical system with the following Hamiltonian:
H2 = A0r + F (0,w,^0)r, (B.1)
where the function F(ф, t, ф0) is 2^-periodic in both the canonical coordinate ф and the independent variable w. It is analytic in the small parameter ф0, that is,
F = фо^(ф, t) + ^2(ф, t) + ... (B.2)
By the canonical change of variables ф, r ^ 9, p the Hamiltonian (B.1) can be reduced to the following normal form:
K2 = Xr, A = A0 + ^A + ф0 A2 + ... (B.3)
The above change of variables can be defined by a generating function of the form
ЗД, P, t, Фо) = ФР + №ЛФ, t)p + ФоЗД, t)p + ... (B.4)
The algorithm for constructing the generating function (B.4) includes the following steps: 1. In the first step we calculate A1, Ф1 and Ф1 as
2n 2n
0 0
2п 2П
J j Fl(ф, t) ctydt, (B.5)
* 1 t) = - j(F(Aoi + t) - Ai) dt, (B.6)
$i(<M) = *i(0 - Aot,t). (B.7)
Here the arbitrary integration function of ^ should be chosen in such a way as to ensure the 2^-periodicity of the function t).
ra"1 дФ„_ !
2. In the nth step we calculate
Gn(0,t) = Fn + J2Ft^±, (B.8) i=1
2n 2n
J J Gn($,t) dtydt, (B.9)
4тг2 ' '
0 0
ъп(ф, t) = - J(Gn(Xot + Ф, t) - xn) dt, (B.10)
Фп(ф,г) = ъп(ф - Xot,t). (B.11)
Here also the arbitrary integration function of ф should be chosen in such a way as to ensure the 2^-periodicity of the function Фп(ф, t).
The change of variables that normalizes the Hamiltonian is written in terms of the generating function by the formulae
dS
в = — = ф + фоФ^ф, t) + Ф20ф2(Ф, t) +..., (в.12)
dS ( , дФ1 ,2 дФ2 \ .
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