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7Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 3, pp. 247-261. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210301
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70F15, 70H09, 70H14
On the Dynamics of a Gravitational Dipole
A. P. Markeev
An orbital gravitational dipole is a rectilinear inextensible rod of negligibly small mass which moves in a Newtonian gravitational field and to whose ends two point loads are fastened. The gravitational dipole is mainly designed to produce artificial gravity in a neighborhood of one of the loads. In the nominal operational mode on a circular orbit the gravitational dipole is located along the radius vector of its center of mass relative to the Newtonian center of attraction.
The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a neighborhood of its nominal mode. The orbit of the center of mass is assumed to be circular or elliptic with small eccentricity. Consideration is given both to planar and arbitrary spatial deviations of the gravitational dipole from its position corresponding to the nominal mode. The analysis is based on the classical Lyapunov and Poincare methods and the methods of Kolmogorov-Arnold-Moser (KAM) theory. The necessary calculations are performed using computer algorithms. An analytic representation is given for conditionally periodic oscillations. Special attention is paid to the problem of the existence of periodic motions of the gravitational dipole and their Lyapunov stability, formal stability (stability in an arbitrarily high, but finite, nonlinear approximation) and stability for most (in the sense of Lebesgue measure) initial conditions.
Keywords: nonlinear oscillations, resonance, stability, canonical transformations
Received May 25, 2021 Accepted August 9, 2021
This research was carried out within the framework of the state assignment (registration No. AAAA-A20-120011690138-6) at the Ishlinskii Institute for Problems in Mechanics, RAS, and at the Moscow Aviation Institute (National Research University).
Anatoly P. Markeev anat-markeev@mail.ru
Ishlinsky Institute for Problems in Mechanics RAS pr. Vernadskogo 101-1, Moscow, 119526 Russia Moscow Aviation Institute (National Research University) Volokolamskoe sh., 4, Moscow, 125080 Russia
1. Introduction. Equations of motion and their particular solutions
Let l be the length of the rod of the gravitational dipole and let M and m be the masses of its end loads. Assume that l is negligible compared to the characteristic size of the orbit, so that the motion of points M and m relative to their center of mass O has no influence on the orbit of the center of mass itself [1-3]. We assume the orbit to be elliptic with eccentricity e.
Fig. 1. The orbital and associated coordinate systems. Euler angles.
Let OXYZ be an orbital coordinate system (Fig. 1). The axis OX is directed along the radius vector FO of the center of mass of the gravitational dipole relative to the attracting center F (not shown in Fig. 1), and the axes OY and OZ are parallel, respectively, to the transversal and the normal to the plane of the orbit. The coordinate system Oxyz is rigidly attached to the gravitational dipole. The axis Oz is directed along the rod in the direction from point m to point M. We specify the orientation of the attached coordinate system Oxyz relative to the orbital coordinate system OXYZ by Euler angles p, § and p. For projections of the absolute angular velocity of the rigid trihedron Oxyz onto its axes we have [4] the following expressions:
ux = (ip + V)a31 + § cos p, uy = (ip + ¿>32 - § sin p, uz = (ip + v)a33 + p, a31 = sin § sin p, a32 = sin § cos p, a33 = cos §.
(1.1) (1.2)
The dot in (1.1) denotes the time derivative t, and v is the angular velocity of rotation of the vector FO (v is the true anomaly in the Keplerian motion of the center of mass in the elliptic orbit),
(1 - e2)3/2
(1 + e cos v )2
(1.3)
where w0 is the angular velocity for circular (e = 0) motion of the center of mass.
For the moments of inertia of the gravitational dipole relative to the axes Ox, Oy and Oz we have the expressions
mM 2
A = B =
-12
m + M
C = 0.
(1.4)
The direction cosines an, a12, a13 of the vector FO in the system Oxyz are given by the equations
a11 = cos ^ cos p — sin ^ cos $ sin p,
a12 = — cos ^ sin p — sin ^ cos $ cos p, (1.5)
a13 = sin ^ sin $.
The kinetic energy of the gravitational dipole in its motion relative to the center of mass is calculated by the formula
1
2'
which, with Eqs. (1.1), (1.2) and (1.4) taken into account, is written as
T=-(Aw2x + Bu2y + Cu2z),
T = -A 2
(lp + Ï)2 sin2 § + è2 . (1.6)
The potential energy in the motion of the gravitational dipole relative to the center of mass is given by the equation [1, 2]
3 w2
n = - _°2)3(1 + ecos vfiMi + Ba\2 + Ca2Vi).
Taking expressions (1.4) and (1.5) into account, we can rewrite this equation as
3
n = -- _ 0 (1 +ecos z/)3 sin2 0 sin2 i?. (1.7)
2 (1 — e )
In (1.7), the quantities that are independent of the Euler angles and insignificant for the equations of motion have been neglected.
The equations of motion can be written in the form of a system of ordinary differential fourth-order equations with the Lagrangian L = T — n. The precession and nutation angles ^ and § will be generalized coordinates. But in what follows the Hamiltonian form of the equations of motion will be used. The dimensionless momenta p^ and corresponding to the angles ^ and § will be given by the equations
dL A(jjq dL A(JJq
(l_e2)3/2^' = (1_e2)3/2^'
and the true anomaly v (see (1.3)) will be taken as an independent variable. In the variables §, p^ and the equations of motion of the gravitational dipole relative to its center of mass have the form
# dG d§ _ dG dp^ dG dp# dG
dv dp^ dv dp$ dv d^ dv dè
where [4]
(1.9)
G = —--5- + --^-ÏÔ-P4, ~ -(1 + ecos v) sin2 0 sin2 )?. (1.10)
2(1 + ecos v)2 sin § 2(1 + ecos v)2 1 ^ 2y > v \ >
If the orbit is circular, then Eqs. (1.9) admit the (Jacobi) integral
p2 p2 3
\ , + -f ~ H - o sin2 ^ sin2 § = const. (1.11)
2 sin2 § 2 2
In the case of a circular orbit, Eqs. (1.9) have the particular solution
n n
0 = g ' = 2 ' ^ = V{) =
which corresponds to the equilibrium position of the gravitational dipole in the orbital coordinate system OXYZ when it is fixed along the axis OX. This equilibrium is Lyapunov stable since in its neighborhood the Jacobi integral (1.11) is a positive definite function [1-4].
Fig. 2. A gravidipole in the plane of an elliptical orbit.
In the case of an orbit of arbitrary eccentricity, Eqs. (1.9) admit a family of particular solutions corresponding to motions in which the gravitational dipole always remains in the plane of the orbit (Fig. 2). For this family § = n/2, = 0, and the quantities p and p^ are defined from the equations
-7- = -,-—-rrr — 1, —r- = 3(1 + ecos v) sin 0cos 0. (1-13)
dv (1 + e cos v )2 dv
The motions described by these equations have previously been examined in sufficient detail (see the monographs [1-4] and references therein). In the case of a circular orbit, they describe pendulum motions of the gravitational dipole: oscillations in a neighborhood of stable equilibrium p = ±n/2 with an arbitrary amplitude (not exceeding n/2), unstable equilibrium p = 0 or n (the gravitational dipole lies along the tangent to the orbit), rotations in the plane of the orbit and motions asymptotic to unstable equilibrium.
In an orbit of small eccentricity (0 < e ^ 1), stable equilibrium (1.12) gives rise to the motion p = ; e), p^ = p^(v; e) which is 2n-periodic in v (eccentricity oscillations):
n
i) = ~, p.ff = 0, (1.14)
n 3
ip* = — + esinz/ — -e2 sm'2u + • • • , = 1 + 3ecosz/ + 3e2 sin2 v + • • • . (1-15)
The dots in (1.15) denote convergent series starting with terms of degree 3 and higher in e. It is shown [5] that eccentricity oscillations are stable with respect to perturbations p and p^.
The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a circular or a weakly elliptic orbit in a neighborhood of an equilibrium point or eccentricity oscillations in the presence of arbitrary small perturbations (both those leaving the gravitational dipole in the plane of the orbit and arbitrary spacial perturbations). Special
attention is also given to the nonlinear problem of constructing and investigating the stability of periodic Lyapunov oscillations arising from stable equilibrium, and to the linear problem of the stability of oscillations of the gravitational dipole in a circular orbit under amplitudes that are not small.
Small linear oscillations of the gravitational dipole in a neighborhood of stable equilibrium in a circular orbit are described in detail in the monograph [6]. Regions of linear instability and stability of planar oscillations of the gravitational dipole of large amplitude (close to n/2) in a circular orbit are studied in [7].
2. The Hamiltonian function in a neighborhood of the particular solutions (1.12) and (1.14), (1.15)
To explore the motion of the gravitational dipole in a neighborhood of its equilibrium (1.12) in a circular orbit or in a neighborhood of eccentricity oscillations (1.14), (1.15) in an elliptic orbit, we introduce new canonically conjugate variables Qi, Q2, Pi and P2 by setting
n
ti=2+Q^ = Po = Pi, l^=vl + P2- (2-1)
The new variables correspond to the Hamiltonian function that is calculated by the formula
where G is the function (1.10), in which ft, p, pl9 and p^ have been replaced with their expressions (2.1).
The function (2.2) can be represented by a convergent series in powers of eccentricity e:
$ = + + e2 + O(e3), (2.3)
= ^gf + ~ \ cos2 Qi(l + 3 cos2 Q2), (2.4)
cos V(P22 - sin2 Q1P2 - 3 sin2 Q1 + 1) ~
=--—-^rk-—-- ~ cos vP{+
i cos2 Qi i
33 +- sinz/(sin2Q2 cos2 Qi — 2Q2) — - cos vcos2 Qi cos2 Q2, (2.5)
3cos2 vP22 - 2cos2v sin2 Q1P2 + 2 sin2 v sin2 Q1 3 2 „2
$2 =----o^ -— + - COS^ vP{-
2 2 cos 2Qi 2 1
33 — - sin2z/(sin2Q2 cos2 Qi — 2Q2) + 2 s^n2 ^cos2 Qi cos '2Q2. (2.6)
3. On the methods and algorithms of investigation
The analysis of the motion of the gravitational dipole is carried out using classical and modern analytical and numerical methods of investigating dynamical systems described by differential Hamilton equations [8-21].
To analyze motion near stable equilibrium of the gravitational dipole or its eccentricity oscillations, use is made of the algorithms of the Deprit-Hori method [4, 22]. This method
considers systems with the Hamiltonian function ^ which can be represented by a series in powers of e (0 < e ^ 1):
^ em
* = E u' t3-1)
in.
m=0
where u = (u1, u2) and U = (U1t U2) are vectors of generalized coordinates and momenta (in the problem of the dynamics of the gravitational dipole we consider here they are two-dimensional). A search is performed for the canonical univalent near-identity transformation
u, U — v, V, (3.2)
which transforms the Hamiltonian (3.1) to the new function
S=E^m(v,V>I/). (3.3)
m=0
The generating function W of the transformation is represented as the series
^ em
W= E U' ^ (3-4)
m=0
To calculate the functions xm in the expansion (3.3) and the transformation (3.2), recurrent relations have been obtained in the Deprit-Hori method. We write them out for m = 0, 1 and 2:
Xo = ^o(v, V, v), (3.5)
dW (v V v)
Xi = 0i(v, V, v) + Lrf,0(v, V, v)--' \ (3.6)
X2 = ^2(v, V, v) + (v, V, v) + L1X1 (v, V, v) + dW (v V v)
+L20o(v, V, v)--' (LmR = (R, Wm)). (3.7)
The change of variables (3.2) is expressed in terms of the functions Wm(v, V, v) by the formulae
In (3.6)-(3.8), (f, g) denotes the Poisson bracket of the functions of v, V and v:
(f ) = V ( df dg df dg\
In the problem of the motion of the gravitational dipole the functions W1 and W2, which are defined from Eqs. (3.6) and (3.7), either do not depend on v (in the case of a circular orbit) or are 2^-periodic in v (when the orbit is elliptic). They are chosen so as to simplify as much as possible the functions x1 and x2 in the transformed Hamiltonian function
S = Xo + tXi + i ~ \2 + 0(e3). (3.9)
4. A circular orbit
In the case of a circular orbit the motion of the gravitational dipole relative to its center of mass is described by equations given by the Hamiltonian function $0 from (2.4). Its series expansion contains no terms linear in Q1, Q2, P1, P2. Neglecting the constant to which the function $0 is equal in the stable equilibrium position (1.12) and introducing, instead of Q1, Q2, P1, P2, new canonically conjugate variables xx, x2, X1, X2 by the formulae
Qi = ^x» $2 = ^74*2, Pi = V2X,, P2 = 3^x2, (4.1)
we find that the new variables correspond to the Hamiltonian function r which can be represented by the convergent series
r = r2 + r3 +r4 + ... , (4.2)
where rfc is a form of degree k in Xi, X2, Xi, X2, and
T2 = -^(xj + Xl) + + X22), (4.3)
T3 = ^-xfX2} r4 = -±xi - fxfxi + ^xW - ±xi (4.4)
In (4.3), a1 and a2 are frequencies of small (linear) oscillations of the gravitational dipole in a neighborhood of its stable equilibrium (1.12) in the circular orbit:
<7! =2, a2 = A/3. (4.5)
Conditionally periodic motions. The quantity k1a1 + k2a2 cannot vanish for any integers k1 and k2 that are nonzero simultaneously. Consequently, the problem of nonlinear oscillations near the equilibrium (1.12) is nonresonant.
We reduce the function r to its normal Birkhoff form in terms through degree four. To do so, we use the Deprit-Hori method described above. It is more convenient to perform calculations in complex conjugate variables rather than real variables. To this end we make the canonical transformation Xj, Xj ^ Uj, Uj (j = 1, 2)
Uj = Xj + iXj, Uj = Xj — iXj (i being an imaginary unit). (4.6)
The transformed Hamiltonian function is ^ = 2ir, where r is the function (4.2), in which the variables Xj and Xj are expressed in terms of Uj and Uj in accordance with the transformation (4.6). If we assume that the quantities Xj and Xj are of order e, then the function ^ can be written as the series (3.1), in which the functions do not depend on v, and
p0 = ia1 u1 U1 + ia2u2U2, = 2ir3, p2 = 4ir4. (4.7)
In the last two equations, r3 and r4 are expressed in terms of u1, u2, U1, U2.
The transformation (3.2) with the generating function (3.4) reduces the function ^ to the function (3.3), which does not depend on v. If we then pass from the complex conjugate variables to real coordinates and momenta Zj and Zj by the formulae
= ¿(0; - = + (j = 1,2), (4.8)
then we obtain the normal form F of the Hamiltonian function in real variables: F = s/(2i), where in the function 2 the variables Vj and Vj are expressed in terms of Zj and Z- by formulae (4.8).
The above changes of variables Xj, Xj — Uj, Uj — Vj, Vj — Zj, Zj were made using computer algorithms of analytical calculations. It is shown that the variables Xj and Xj are expressed in terms of Zj and Zj by the following formulae:
2 1/4 „ 8 0/4 ^ 617 o 425 r o
as =z,--31/4^iZ2 + —3i/4z2Zi +--2? + --V3 ziz$+
1 1 13 1 2 39 2 1 16 224 1 4056 1 2
223 153 7
+ 5408*^ - + IS*2'* + (4'9)
2 „i /„ „ 19 2 1 ^ „ „ . 5
Xo = Zo--31/4^iZi - —Z?z2 - -V3z,Z,Zo + —V3zl+
2 2 13 1 1 676 1 2 4 1 1 2 144 2
75 1
+ 3^8-2^1 + Jq^zI + 04, (4.10)
Xl = Zl~ ig33'4*1*2 + h//iZlZ'2 ~ + bZlZ2Z2~
745 ^ 245 , 185 /- ,
4056 2 1 5408 1 1352 1 2 4' v '
X2 = Z2- ^3 3/4z2 - — 33^4 Z\ + ^-z\Z2 - -y/Sz^Z,-2 78 1 39 1 676 1 2 52 1 2 1
_ Ps/3z22Z2 - —Z\Z2 - 1 x/3Z23 + 04. (4.12)
48 2 2 338 1 2 48
In (4.9)-(4.12), O4 denotes the set of terms of degree higher than three in z1, z2, Z1, Z2.
If we now pass from the variables Zj and Zj to symplectic polar coordinates Vj and Pj by the formulae
ZJ = fasin(^i' = (j = 1, 2), (4.13) then the normalized Hamiltonian function F takes the form
F = H + O ((v1 + V2)5/2) , (4.14)
H = + a2r2 - —rj - —rlr2 - -r22. (4.15)
9 2 3 1
—rf--r\ r0--
52 1 13 1 2 4
The Hamiltonian function (4.15) is nondegenerate since
, ( d2H V 105 , .
det fed = 676 (4'16)
V 1 j / i,j=1
Therefore, according to KAM theory [11-15], in a neighborhood of the equilibrium (1.12) there exist invariant tori that are close to tori of a linearized system. The motion of the gravitational dipole which corresponds to these tori is conditionally periodic with frequencies w1 and w2:
dH 9 73 dH x/3 1 ..
UJ i = —- = <7i--fi--r9, UJ9 = —- = (7 2--fi--r9. (4.17)
1 ch\ 1 26 1 13 2' 2 dr2 2 13 1 2 2 v ;
2
For the trajectories Qj (v) and Pj(v) starting in the small neighborhood ^ [Q2 + Pj2] < i of the
j=1
equilibrium (1.12), the relative measure of the initial conditions corresponding to conditionally periodic motions of the gravitational dipole is no less than 1 — O(p1/4).
On periodic Lyapunov motions. Since the ratio of frequencies (4.5) is not an integer, it follows by the Lyapunov theorem of a holomorphic integral [9] that there exist families of short-periodic and long-periodic (in v) motions of the gravitational dipole with periods close to 2n/a1 and 2n/a2.
1. We first consider short-periodic motions. Using the canonical transformation analytic in Zj, Zj (j = 1, 2), from the expansion of the function F(z1, z2, Z1, Z2) as a series (see the previous section) one can eliminate terms linear in z2 and Z2. And then in the transformed Hamiltonian function (4.14) the quantity O ((r1 + r2)5/2), which can be represented by a series
in powers of rj/2 and r1/2, will contain no terms linear in r^2, and the equations of motion admit particular solutions corresponding to short-periodic oscillations of the gravitational dipole:
z1 = e sin p1, Z1 = e cos p1, z2 = 0, Z2 = 0 (0 <e ^ 1), (4.18)
9
^ = r + ^(0), r = iV, Q1 = 2--t~2 + 0(t~4). (4.19)
Here Q1 is the frequency u1 from (4.17), which has been calculated for the values r1 = e2/2 and r2 = 0 corresponding to (4.18).
It follows from (4.1) and (4.9)-(4.12) that in the variables Q1, Q2, P1 and P2 (see (2.1)) the family of solutions (4.18) can be written as
Q i = t— sint^ +g332 (630 sin y?! + 13 sin 3tpi) + 0(t4), Q2 = -t~2^sin2^ + 0(t~4),
r ■ a/2
Pi = tv2 cos — t3——(258 cos — 13 cos 3(/?i) + 0( t4), 1 1 5408 1 1
P2 = —12-^-(13 + 3cos2(/?1) + 0(t4). (4.20)
52
To investigate the orbital stability of periodic motions (4.20), we introduce perturbations n1 and n2 by setting
1 2 I
rl = 2S +i?1' f2 = i?2-
Substituting these values of r1 and r2 into the function (4.14) and passing to the new independent variable t, we obtain a Hamiltonian function of perturbed motion in the form
F = n1 + + C2cn2 + C11 nn + C02ni + O ((|m| + %)5/2), (4.21)
where Set
D = C20A2 - cnX + C02. (4.22)
The condition D = 0 of the isoenergetic nondegeneracy of the function (4.21) is a sufficient condition for orbital stability [13-16]. In the case we consider here,
416 43 264 V
therefore, at sufficiently small e the short-periodic motions of the gravitational dipole are orbitally stable.
2. The family of long-periodic motions is treated in a similar way. For this family,
z1 = 0, Z1 = 0, z2 = e sin Z2 = e cos p2,
^2 = r + ^2(0), T = a2u, a2 = s/3-L;2+ o(£4).
(4.23)
(4.24)
In the variables Qj and Pj we have
Qi = 0, Pi = 0,
33/4
,31/4
Q2 = £— sin (^2 + t —(6 sin (^2 + sin 3(^2) + 0{£ ),
33/4
P2 = Î31/4 COS Lp2 ~ £i~T^(2cos ¥>2 ~~ COS3(/?2) + 0(t4).
48
(4.25)
The long-periodic motions are oscillations of the gravitational dipole in the plane of the orbit.
In analyzing their orbital stability we obtain a Hamiltonian function of perturbed motion in the form (4.21), such that
5
A = T+3r2+0(s<)-
1
73
V^ 1 2 , rM
3
cn = ~Tô ~ + 0(e% C02 = " + 0(t-4).
13 156 From this and Eq. (4.22) we find
52
208
D = —
468
505.e2 + 0(t-4})
8112
therefore, at small e the planar oscillations of the gravitational dipole are orbitally stable.
Planar oscillations of arbitrary amplitude. Consider the problem of the stability of planar oscillations without assuming their amplitudes to be small. We restrict ourselves to analysis in the first (linear) approximation relative to small spatial perturbations.
For ease of calculations we make in the Hamiltonian function (2.4), which defines the equations of motion of the gravitational dipole in a circular orbit, a canonical (with valence 4\/3/3) change of variables Q1, P1, Q2, P2 ^ Q, P, q, p by the formulae
1
1
Qi = -;Q, Pi = ^rP, Q2 = -;Q, P
V3
2
2
2
2
-p
and, instead of v, we take the quantity u = \piv as an independent variable. The new variables correspond to the function $0 which can be represented by the convergent series
$0 = + ^2 +
(4.26)
Here
^0 = 2^ -cos<?, ^2 = 2P +2^
f = ^P + —P+ 2 cosq+ g'
(4.27)
(4.28)
The dots in (4.26) denote a series in powers of Q which starts with terms of degree four, the coefficients of the series being functions of q and p.
The planar oscillations of the gravitational dipole under study are described by equations with the Hamiltonian function p0. They are pendulum-like oscillations that are expressed in terms of elliptic Jacobi functions. Let q = 0 at the initial time, and let a be the amplitude of oscillations (0 < a <n). Then
q = 2arcsin[k sn(u, k)], p = 2k cn(u, k) (k = sin(a/2)). (4.29)
Under the oscillations (4.29) the function p0 is constant,
p0 = 2k2 — 1. (4.30)
The linearized equations of perturbed motion are given by the function p2. If we take the quantity w = nu/(2K(k)) as an independent variable, then these equations can be written as
n = *mP & = j*mfQ. (4.31)
dw n dw n
Here f is the function (4.28). It is 2n-periodic in w and, with Eqs. (4.29) and (4.30) taken into account, can be represented as
/ = 2k2 cn2 (j^-w, kj + '^kcn (j^-w, kj - k2 + 1. (4.32)
If k = 0, the function f is equal to 4/3. Since 4/3 is not equal to the square of a half-integer, parametric resonance is impossible at small k and stability [8] takes place. This conclusion is in agreement with the previous section, where the stability of small planar oscillations of small amplitude is shown not only in the linearized problem of stability, but also in its rigorous nonlinear version.
Under amplitudes that are not small, stability was investigated using numerical calculations. To this end, a calculation was made of the coefficient a depending on k in the characteristic equation of the monodromy matrix of the system (4.31)
q2 — 2ag + 1 = 0. (4.33)
If |a| > 1, then the oscillations are unstable (in the rigorous version of the stability problem), and if |a| < 1, then linear stability [8] takes place.
A graph of the function a(k) is presented in Fig. 3 for values of k from the interval (0, 0.99). If 0 <k <k* = 0.84860807..., then —1 <a < 0.563638594.... The value k = k* corresponds
(see (4.29)) to the value a = a* = 2.026697153____Thus, oscillations of the gravitational dipole
with an amplitude not exceeding Q2 = a*/2 — 58° are linearly stable.
Under amplitudes lying in the interval Q2 < Q2 < Q** — 75°, which correspond to the values of k from the narrow interval
k* <k<k** = 0.968158697... , (4.34)
the inequality a < —1 holds. Here the planar oscillations of the gravitational dipole are unstable. When k > k**, the first instability interval (4.34) will be followed by the stability interval in which —1 < a < 1, and then by the second instability interval in which a > 1:
0.992381028 ... <k< 0.997869232 ... (4.35) _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2021, 17(3), 247-261_"j^
Fig. 3. Graph of the function a(k) for 0 < k < 0.99.
As k increases further, the intervals of linear stability and intervals of instability alternate with each other and become increasingly narrow. For example, the three instability intervals following the interval (4.35) are as follows:
0.9994 ... <k < 0.9998 ... , 0.99996 ... <k< 0.99999 ... , 0.999997 <k < 0.999999 ...
No calculations have been made for values of k that are larger than 0.999999.
5. The elliptic orbit of small eccentricity
Let us make a transformation Q-, P- — y-, Y- in the Hamiltonian function (2.3) using formulae similar to (4.1):
Qi = ^Vi, Q2 = ^V2, Pi = V2Yu P2 = 31/4F2 (5.1)
and expand it as a series in powers of y- and Y- through terms of degree four in y- and Y-. The coefficients of the series will be functions of e and v. Assuming y- and Y- to be quantities of order e and using the Deprit-Hori method, we construct a 2^-periodic change of variables y-, Y- — z-, Z- which reduces the Hamiltonian function to its normal form. The calculations involved will be more cumbersome than those in the case of a circular orbit, since the change of variables will explicitly depend on v. Computer calculations have shown that the normalizing change of variables can be represented as
y3 = x- + S-, Y- = X- + A- (j = 1, 2), (5.2)
where x- and X- are given by Eqs. (4.9)-(4.12), in which O4 is the set of terms of degree higher that three in e, and S- and A- are calculated by the following formulae:
Si = e
sin, (§33/V~2 + - cos, - +
+
+e
K ^ 1
-(1 + 3cos2u)z1 — - sin 2vZx
(5.3)
¿2 = e
sin v
++a**+if31/4Z?+£•*«) -
— cos V
e
27
A1 = e
, . :— cos2z/ | Zo ——VSsm2i'Z9 11 2 44 2
sini/ , iZl - + -^4z2Z1 ) +
(5.4)
2
143
143
+ cos v(zl + + ¿31/4Z1Z2
+e2
+
A
2=e 9
1 1
- sin 2vzi 4 1 + 4
7 r 68
IT ~ 429'
(5.5)
sinz/ (L^oZ _ ™3wZlZl - -33/*z.2z.2) + V11 429 1 1 13 2 J
+ cosu ( + i^33/4z2 + - i^33/4Z2 + -3 ^z2
1 11 2 429 1 13 2 429 1 13 2
e
1 3
— Vdsm'lvzo +-(1 - 99 cos 2u) Z.2
44 2 484 v ' 2
If we introduce symplectic polar coordinates by formulae similar to (4.13):
zi = T? \fir~j sin <Pj' ZJ = 7?"1 \fer~j cos <Pj (i = 1' 2)'
(5.6)
(5.7)
Yi
2 + e 2^2
2\ 1/4
Y2
22 + 15e2\ 1/4 22 - 31e2 J
then the normalized Hamiltonian function can be written as F = H+O(e5), where H is calculated by formula (4.15), in which a1 and a2 have been replaced with A1 and X2 which are functions of e:
Ai = \/4 — e4,
1
A2 = — V3(22 + 15e2)(22-31e2).
(5.8)
1. The part of the function H that is quadratic in V1 and V2 is sign definite. Therefore, for a system with the Hamiltonian function F there exists a sign-definite integral that can be represented by a (possible divergent) series, i.e., the formal stability [17-19] of equilibrium Q- = = Pj = 0 of the system with the Hamiltonian function (2.3) takes place. This implies that the eccentricity oscillations of the gravitational dipole in the presence of spatial perturbations will be Lyapunov stable when terms up to an arbitrarily high (finite) degree in Qj, Pj (j = 1,2) are taken into account in the series expansion of the function (2.3).
2. Using KAM theory, these conclusions on stability can be significantly supplemented and
reinforced. To do so, we note that the Hamiltonian function H is nondegenerate (see (4.16)).
Therefore [11-16], eccentricity oscillations in the presence of spatial perturbations are metrically
stable. For all values of v, most trajectories of the system with the Hamiltonian function F are
described by conditionally periodic functions with three frequencies w1 = dH/dvx, w2 = dH/dv2
and w3 = 1. The set of initial conditions which do not correspond to these trajectories has a relative measure tending to zero as eccentricity decreases.
Trajectories starting in this set of small measure can move far away from their initial points. But this possible motion away from the origin will be extremely slow. This follows from the fact that, since the part of the function H that is quadratic in r1 and r2 is sign definite, it satisfies what is called the steepness conditions [20]. And if one sets r1 (0) + r2 (0) = e ~ e3, then it can
2
be shown [20, 21] that there exist positive numbers c and d such that ^\rj(v) — rj(0)| < ec
j
for 0 < v < exp(e-d).
References
[1] Beletskii, V. V., Motion of an Artificial Satellite about Its Center of Mass, Jerusalem: Israel Program for Scientific Translations, 1966.
[2] Beletskii, V. V., Satellite's Motion about Center of Mass in a Gravitational Field, Moscow: MGU, 1975 (Russian).
[3] Sarychev, V. A., Problems of Orientation of Satellites, Itogi Nauki Tekh. Ser. Issled. Kosmich. Prostr., vol. 11, Moscow: VINITI, 1978 (Russian).
[4] Markeev, A. P., Linear Hamiltonian Systems and Some Problems on Stability of Motion of a Satellite about Its Center of Mass, Izhevsk: R&C Dynamics, Institute of Computer Science, 2009 (Russian).
[5] Zlatoustov, V. A. and Markeev, A. P., Stability of Planar Oscillations of a Satellite in an Elliptic Orbit, Celestial Mech, 1973, vol.7, no. 1, pp. 31-45.
[6] Smirnov, A. S. and Smolnikov, B.A., Spherical Pendulum Mechanics, St. Petersburg: SPbGPU, 2019 (Russian).
[7] Sidorenko, V. V. and Neishtadt, A. I., Investigation of Stability of Long-Periodic Planar Motion of a Satellite in a Circular Orbit, Cosmic Research, 2000, vol. 38, no. 3, pp. 289-303; see also: Kos-micheskie Issledovaniya, 2000, vol.38, no.3, pp. 307-321.
[8] Malkin, I.G., Theory of Stability of Motion, Ann Arbor, Mich.: Univ. of Michigan, 1958.
[9] Malkin, I.G., Some Problems in the Theory of Nonlinear Oscillations: In 2 Vols.: Vol. 1, German-town, Md.: United States Atomic Energy Commission, Technical Information Service, 1959.
[10] Birkhoff, G.D., Dynamical Systems, AMS Coll. Publ., vol.9, Providence, RI: AMS, 1966.
[11] Kolmogorov, A. N., Preservation of Conditionally Periodic Movements with Small Change in the Hamilton Function, in Stochastic Behaviour in Classical and Quantum Hamiltonian Systems (Volta Memorial Conference, Como, 1977), G. Casati, J.Ford (Eds.), Lect. Notes Phys. Monogr., vol. 93, Berlin: Springer, 1979, pp. 51-56; see also: Dokl. Akad. Nauk SSSR (N. S.), 1954, vol. 98, no. 4, pp. 527-530.
[12] Arnol'd, V. I., Proof of a Theorem of A. N. Kolmogorov on the Invariance of Quasi-Periodic Motions under Small Perturbations of the Hamiltonian, Russian Math. Surveys, 1963, vol. 18, no. 5, pp. 9-36; see also: Uspekhi Mat. Nauk, 1963, vol. 18, no. 5, pp. 13-40.
[13] Arnol'd, V. I., Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics, Russian Math. Surveys, 1963, vol.18, no. 6, pp. 85-191; see also: Uspekhi Mat. Nauk, 1963, vol. 18, no. 6(114), pp. 91-192.
[14] Arnol'd, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1997.
[15] Arnol'd, V.I., Kozlov, V. V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol.3, Berlin: Springer, 2006.
[16] Moser, J.K., Lectures on Hamiltonian Systems, Mem. Amer. Math. Soc., No. 81, Providence, R.I.: AMS, 1968.
[17] Moser, J., New Aspects in the Theory of Stability of Hamiltonian Systems, Comm. Pure Appl. Math, 1958, vol. 11, no. 1, pp. 81-114.
[18] Moser, J., Stabilitätsverhalten Kanonischer Differentialgleichungssysteme, Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II, 1955, vol. 1955, no. 6, pp. 87-120.
[19] Glimm, J., Formal Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1964, vol. 17, no. 4, pp. 509-526.
[20] Nekhoroshev, N. N., Behavior of Hamiltonian Systems Close to Integrable, Funct. Anal. Appl., 1971, vol. 5, no. 4, pp. 338-339; see also: Funkts. Anal. Prilozh., 1971, vol.5, no. 4, pp. 82-83.
[21] Nekhoroshev, N. N., An Exponential Estimate of the Time of Stability of Nearly Integrable Hamiltonian Systems, Russian Math. Surveys, 1977, vol. 32, no. 6, pp. 1-65; see also: Uspekhi Mat. Nauk, 1977, vol.32, no.6(198), pp. 5-66, 287.
[22] Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Appl. Math. Sci., vol.8, New York: Springer, 1972.
