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10Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 4, pp. 453-464. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210407
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 34D20, 37J40, 70K30, 70K45, 37N05
On the Orbital Stability of Pendulum-like Oscillations of a Heavy Rigid Body with a Fixed Point in the Bobylev - Steklov Case
B. S. Bardin, E. A. Chekina
The orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev - Steklov case is investigated. In particular, a nonlinear study of the orbital stability is performed for the so-called case of degeneracy, where it is necessary to take into account terms of order six in the Hamiltonian expansion in a neighborhood of the unperturbed periodic orbit.
Keywords: rigid body, rotations, oscillations, orbital stability, Hamiltonian system, local coordinates, normal form
1. Introduction
The modern methods of Hamiltonian dynamics allow rigorous conclusions about the stability of periodic motions for a wide class of problems in classical and celestial mechanics. The classical problem of stability of periodic motions of a heavy rigid body with one fixed point belongs to such a class of problems. Suppose the center of mass of the body lies in the plane of the two main axes of inertia. In this case, there exist plane periodic motions such that the third axis of the ellipsoid of inertia retains a constant horizontal position in absolute space, and the body performs periodic pendulum-like oscillations or rotations relative to this axis. These periodic
Received December 07, 2021 Accepted December 15, 2021
This work was supported by the grant of the Russian Scientific Foundation (project No. 19-11-00116) at the Moscow Aviation Institute (National Research University).
Boris S. Bardin [email protected]
Moscow Aviation Institute (National Research University) Volokolamskoe sh. 4, Moscow, 125993 Russia
Mechanical Engineering Research Institute of the Russian Academy of Sciences M. Kharitonyevskiy per. 4, Moscow, 101990 Russia
Moscow Automobile and Road Construction State Technical University (MADI) Leningradsky pr. 64, Moscow, 125319 Russia
Evgeniya A. Chekina [email protected]
Moscow Aviation Institute (National Research University) Volokolamskoe sh. 4, Moscow, 125993 Russia
motions will obviously be unstable along the angular coordinate (the angle of deviation from the vertical). However, from the theoretical point of view, as well as for applications, the problem of the orbital stability of such periodic motions is of considerable interest.
The orbital stability of pendulum-like motions of a heavy rigid body with a fixed point has been studied in many papers. In integrable cases, the orbital stability of periodic motions can be investigated using topological methods [1]. In some partial cases, it was possible to apply the method of Lyapunov functions [2, 3]. The effect of the presence of a rotor on pendulum-like motions of a heavy rigid body has been investigated in [4].
Since, the equations of motion of a heavy rigid body with a fixed point can be written in the form of Hamiltonian equations, the general approach developed for the investigation of periodic trajectories of Hamiltonian systems can be applied for the study of orbital stability. In accordance with this approach, it is necessary to introduce the so-called local coordinates [5] in a neighborhood of a periodic orbit and write perturbed motion equations. Rigorous stability analysis of the perturbed system thus obtained can be carried out using the well-developed methods of normal forms [6, 7] and KAM theory [8, 9].
In specific dynamics problems, introducing local coordinates in a neighborhood of periodic motions, and obtaining equations of perturbed motion in explicit form can turn out to be a difficult problem. In the problem of the orbital stability of pendulum-like periodic motions local coordinates can be introduced by transformation to action-angle variables in the region of periodic motion. This approach allows one to perform a nonlinear stability analysis and obtain rigorous conclusions on orbital stability in both integrable and nonintegrable cases [10-14]. However, in using this approach, one can generally encounter some technical difficulties in calculating explicit expressions for the coefficients at higher-order terms of the series expansion of a Hamiltonian in a neighborhood of unperturbed periodic motion.
Another method for introducing local coordinates was proposed in [15]. In accordance with this method, local coordinates can be introduced by a nonlinear canonical change of variables, which is constructed in the form of power series with respect to a new variable (local coordinate). It makes it possible to avoid the above-mentioned technical difficulties and to obtain an explicit form of the series expansion of a Hamiltonian of perturbed motion up to 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 [15, 16]. In particular, it provides a way to study the orbital stability of pendulum-like oscillations of the body with respect to its largest principal axis of inertia in the special case where a degeneracy of fourth-order terms in the series expansion of a Hamiltonian of perturbed motion takes place.
In this paper, we also consider the orbital stability of pendulum-like oscillations in the Bobylev-Steklov case. Using the approach developed in [15], we study the orbital stability of pendulum-like oscillations of the body with respect to its smallest or middle principal axis of inertia in the above-mentioned special case of degeneracy.
2. Problem statement
Consider the motion of a rigid body about a fixed point O in a uniform field of gravity. To define the body's position, we use the following coordinate systems. OXYZ is a fixed coordinate system whose axis OZ is directed vertically upward. Oxyz is a relative coordinate system that is rigidly attached to the body. It is formed by the principal axes of inertia of the body for point O. We use the standard notation denoting by A, B and C the moments of inertia corresponding to the axes of the relative coordinate system Oxyz.
In what follows it is assumed that the body's center of mass lies on the axis Ox at distance l from point O, and the moments of inertia satisfy the relations A = 2B. That is, the so-called Bobylev-Steklov case takes place (see, e.g., [17]).
Let us specify the orientation of the body in the fixed coordinate system OXYZ by using the Euler angles 0, p. By introducing generalized momenta p^, pg, pv corresponding to the Euler angles 0, p, one can write the equations of motion in Hamiltonian form. The angle ^ is a cyclic coordinate, which is why the corresponding momentum p^ is a first integral that is equal to zero on the unperturbed pendulum-like periodic motions. We also put p^ = 0 for the perturbed motion.
Now we introduce the dimensionless time t = ¡t, as well as dimensionless coordinates q1, q2 and momenta p1, p2 by means of the following formulae:
where ¿t = \Jrngl/C; mg is the weight of the body.
In variables qi, pi (i = 1, 2) the motion of the body is described by the canonical equations
with the Hamiltonian function
H = i { [2 + a( 1 + sin2 q{) tan2 q2] pf + cxPip.2 sin(2q{) tan q2 + a(2 — sin2 ^1)^2} —
— cos q1 cos q2, (2.3)
where a = C/B (1 < a < 3).
The system (2.2) has a one-parameter family of periodic solutions of the form
qi = f (t ), pi = g(T), q2 = p2 = 0, (2.4)
where the functions f (t), g(T) are a solution of the canonical system with the Hamiltonian H(0) = = 1/2p2 — cos q1. The solution (2.4) describes planar pendulum-like motions of the body relative to the fixed axis of inertia Oz. Depending on the value of the constant h of the energy integral
H (0 = h, the planar motions are either asymptotic (h = 1) to the unstable equilibrium position p = n/2 of a rigid body or are periodic motions: oscillations (|h| < 1) in a neighborhood of the stable equilibrium position p = 3n/2 or rotations (h > 1) relative to the axis Oz, which keeps a fixed horizontal position.
In [11] the problem of the orbital stability of the above pendulum-like periodic motions has been studied for most values of parameters a and h. In this paper we study the previously unconsidered case of degeneracy where the terms of order six in the expansion of a Hamiltonian in a neighborhood of a periodic orbit have been taken into account to solve the problem of orbital stability.
3. Local coordinates and equations of perturbed motion
To study the orbital stability of pendulum-like periodic motions, we perform a canonical change of variables q1, p1, q2, p2 n, q2,p2
qi = f(t,n), p1 = o(c,v), (3.1)
such that in the new variables the family of periodic solutions (2.4) has the form
£ = t + £ (0), n = q2 = p2 = 0. (3.2)
The problem of the orbital stability of the periodic solution (3.2) is equivalent to the problem of Lyapunov stability with respect to the variables n, q2, P2, which we will call local coordinates in what follows.
The local coordinates can be introduced in different ways. In [15] it was proposed to construct the canonical change of variables (3.1) in the form of the following power series:
F(0 n) = f (£) + a! (On + a2(e)n2 + a3(0n3 + •••, n) = g(0 + )n + b2(On2 + b3(On3 + •••,
where the periodic coefficients ai((), bi(() can be calculated by the formulae
... 1 dg 1 d2f 1 d3g W2 dg
MO = -772-77' MO = TTFTITTTI > MO =
(3-3)
V2 dO 2V4 d£2' 3Vsy 6V6 d£3 3 V8 d0
1 df _ 1 d2g ,,.. 1 d3f W2 df (0
(3^4)
V2 dC 2 2V4 d(2' 3Vsy 6V6 d(3 3 V8 d( '
where
^(fHl)'-
We recall that the functions f (0, g(0 are a solution (see (2.4)) of the canonical system with the Hamiltonian H (°) describing unperturbed pendulum-like periodic motion of the rigid body. That is why the following relations take place:
I=* !=-*". <°«>
d2 f ■ f d2g
d3/ d3g
w = ~9COSf> W
ZTP7 = -g cos/, ^ = sin/(cos/ + #2). (3.8)
Taking Eqs. (3.6)-(3.8) into account, we obtain the following explicit form of the canonical change of variables (3.3):
sin f sin f 2 . <71 = / + -^?-^? +
. g g cos f 2 . Pi = s + ^ -+
sin f (cos2 f + g2) W2 sin f
6Ü6 + 3F8
n3 + 0(n4 ),
g cos f W2g
n3 + 0(n4 ),
M
where V2 = g2 + sin2 f and W2 = g2 cos2 f + sin2 f.
The argument of the functions f and g in (3.9) is the new canonical variable 0 The explicit expressions for f and g depend on the type of periodic motion. In the case of oscillations (when \h\ < 1) they are given by the formulae [10]:
2 h + 1
f = 2arcsin[fc1sn(0 hj], g = cn(0 kx), = ——, (3.10)
and in the case of rotations (when h > 1) by the formulae
2
/ = 2 am(£, k2), g = 2k2l dn(£, k2), k2 = (3.11)
In (3.10)—(3.11), standard notation is used for the elliptic functions [18].
The period of pendulum-like motions is equal to 2п/ш, where ш = n/(2K(kl)) in the case of oscillations and ш = n/(k2K(k2)) in the case of rotations. K denotes the complete elliptic integral of the first kind.
By substituting (3.9) into (2.3), we obtain the following Hamiltonian of the system of equations of perturbed motion in a neighborhood of the periodic orbit:
г = r + r4 + ... + + ..., (3.12)
where Г2т is a form of degree 2m in q2, p2, n1/2. The forms Г2, Г4 and Г6 of the expansion (3.12), which are required for further analysis, read
Г2 = n + $2ofe,P2, С ), (3.13)
Г4 = Х4ЮП2 + $22^2, P2, С)n + Ф40^2, P2, С), (3.14)
Гб = Хб(С)n3 + ^24(q2, P2, С)n2 + Ф42(q2, P2, C)n + $6ote, P2, С), (3.15)
where
фтк(q2,P2,C)= E p2, m = 2,4,6. (3.16)
i+j=m
The coefficients х4(С), Х6(С) and those coefficients of the forms Фтк which are not equal to zero read
X4(0 = ¿4 (cos / - l)(sin2 / - g2), xe(0 = g^î(1 " cos /)2, = [(sin2 / cos / - cos2 / + 2)ag2 - sin2 /], ¥>ff(0 = sin /[sin2 / — cos /(cos / + 1)],
= a Sin2/cos/, 420}({) = ±[a(2 - cos2 f)g2 + 2 cos /],
fn(0 = sin /cos/, = -^-(cos2 / + 1),
(ow 1
V>40(0 = g
a(2 - cos2 f)g2 - ^ cos /
> 4°i}(e) = ^sm/cos/,
4o} (0 = ¿4(1 - cos f)(ag2(2 cos /(cos / + l)2 + 1) + sin2 /), ^(n(e) = ^4«i/sin3/(4cos/ + l), P02 (0 = ^4«sin2 /(! " cos /)(2 cos / + 1), = [8«i/2(sin2 /(cos / + 1) + 1) + sin2 /],
(3.17)
42i}(0 = ^ja sin/(cos / + l)(2cos / - 1), = y^[68a^2(sin2/+ 1) +cos/], = -^ag sin /cos/,
where f, g are the functions defined by formulae (3.10) in the case of oscillations and by formulae (3.11) in the case of rotations.
By virtue of the equations of motion with the Hamiltonian (3.12), the coordinate £ is an increasing function of the variable t. Therefore, in the problem of motion stability, this coordinate can play the role of time. To describe the motion on the zero isoenergetic level, we take the coordinate £ to be a new independent variable. In addition, from the equation r = 0 with small q2, p2, n we have n = —y(q2, P2, £). The function Y(q2, p2, £) is the series
Y = Y2 + Y4 + ... + Yk + ..., (3.18)
where Yk is a form of degree k in q2, p2 with coefficients periodically depending on £. The forms y2 , Y4 and y6 have the following explicit form:
Y2(q2> P2, £) = $2°^ P2, £), (3.19)
Y4(q2 , P2, £) = X4(£ )Y2(q2) P2, £) — Y2 ^2, P2, 0^22^2, P2, £) + $4° ^2, P2, £), (3.20)
Y6(q2, P2, £) = —X6(£b3^ P2, £) + Yi^ P2, £)$24(q2, P2, £) +
+ 2X4(£)Y2(q2, P2, £)Y4(q2, P2, £) — Y2(q2, P2, £)$42(q2, P2, £)— (3.21) — Y4(q2, P2, £)$22(q2, P2, £) + $6°(q2, P2, £).
The equations of motion on the isoenergetic level r = 0 have the canonical form with the Hamiltonian y(q2, P2, £). By introducing the new independent variable w = w£ these equations can be written as follows:
dq2 dK d'P2 dK
dw dp2' dw dq2' (3.22)
where
K{q2, p2, w) = -r/{q2, P2, u~lw). (3.23)
w
The new Hamiltonian K is T-periodic in w, with T = n in the case of oscillations and T = 2n in the case of rotations.
Thus, the problem of the stability of pendulum-like periodic motions of a rigid body reduces to investigating the stability of the equilibrium point q2 = P2 = 0 of the reduced system (3.22).
4. A study of the orbital stability of pendulum-like oscillations in the case of degeneracy
The problem of the stability of pendulum-like periodic motions of a heavy rigid body in the Bobylev-Steklov case has been considered previously. In [12, 15, 16], the case of pendulumlike motions relative to the axis Oy was discussed. The case considered here of pendulum-like motions relative to the axis Oz was studied in [11].
In the above-mentioned papers [11, 12], the local coordinates were introduced by using action-angle variables in the region of phase space corresponding to unperturbed periodic motions. In this case, the Hamiltonian of the perturbed motion is obtained in the form of a power series of the perturbation of the action variable and canonical variables q2, P2. In such an approach, the construction of the expansion of a Hamiltonian in the neighborhood of a periodic orbit requires an explicit form of its partial derivatives with respect to the action variable. This leads to rather cumbersome and time-consuming calculations in cases of degeneracy, where solving the problem of orbital stability requires an analysis including terms of order higher than four in the expansion of the Hamiltonian in a neighborhood of the periodic orbit. For this reason, in particular, the above-mentioned cases of degeneracy were not considered in [11, 12].
Fig. 1. Stability diagram for pendulum-like motions of a rigid body in the Bobylev - Steklov case
In [11], nonlinear analysis of the orbital stability of the pendulum-like motions considered here was carried out, taking into account terms up to fourth order in the expansion of the Hamiltonian in a neighborhood of the periodic orbit. We give a brief account of the main results of [11]. In the parameter plane (a, h) an orbital stability diagram, shown in Fig. 1, was obtained. In the hatched regions, the pendulum-like periodic motions are orbitally unstable. In the unhatched regions, orbital stability takes place. An exception can only be the curve £, where a degeneracy takes place. For parameter values on this curve, the problem of orbital stability can be solved using terms of order six, or perhaps of higher orders, in the expansion of the Hamiltonian of the perturbed motion.
We also note that on the boundaries separating the stability and instability regions the study of orbital stability has been also performed in [11]. The parts of the boundaries shown by a solid curve correspond to the orbital stability and parts of the boundaries indicated by a dashed curve correspond to the orbital instability.
Using the above explicit expressions for the Hamiltonian of the perturbed motion, one can investigate the orbital stability of pendulum-like oscillations for parameter values corresponding to the degeneracy curve £.
First, we consider the case of oscillation with small amplitudes. In this case, it is possible to investigate orbital stability analytically by using the small parameter method.
As a small parameter of the problem, we choose the quantity k1 = sin //2 (where /3 is the amplitude of oscillations). This quantity is the modulus of the elliptic functions in the expressions (3.10) and is related to the constant energy h by k2 = (h + 1)/2. In order to investigate the stability of the equilibrium point in the case of degeneracy, it is necessary to
reduce the Hamiltonian (3.23) to normal form up to terms of degree six and to apply the criterion for stability of the Hamiltonian system with one degree of freedom [8, 9].
At the first stage of the normalization procedure, we present the quadratic part of the Hamiltonian (3.23) to the form corresponding to a harmonic oscillator.
Using the well-known expansions of elliptic functions [18], we obtain the following expansion of the quadratic part of the Hamiltonian (3.23)
1 fc2
K2 = t(<72 + aP2) + -j-{(4a cos 2w + 4a + 4 cos 2w — 3)g| + 8aq2p2 sin2w+ 2 8
+ (4acos 2w — 3a)p2} + O(kf). (4.1) Let us pass to polar canonical variables 0, r using the formulae
q2 = a1/4 \/2r sin^, p2 = a-1/4 \/2r cos (f). (4.2)
Now the Hamiltonian quadratic part K2 reads
k'2 r
K2 = s/ar + | s/a(2a - 3) + 2yfa{a + 2) cos 2w - 2a3/2 cos 20+
+ a (2 - y/a) cos(2</> - 2w) - a (2 + s/a) cos(2</> + 2w) J + O(kj). (4.3)
a (2 - vaj cos(20 - 2w) - a r> 1 , •,„„,«. , , ,< iM
By making a canonical near-identity ^-periodic in w change of variables r d, p defined by the generating function
k2 i
S(<p, p, w) = <pp-\—-p < 2 s/a(a + 2) sin 2w — 2a sin 2<p+ 8
+ "3/2 ~ q2 + 2a sin(20 _ 2w) - {a'i/2 2a) sin(20 + 2w) \ +0(kf), (4.4)
a — 1 a — 1
we exclude the dependence of K2 on w and bring the quadratic part of the Hamiltonian into the following form:
K2 = fip, (4.5)
where
n = V^+M'2"~?,)k2 + 0(ki). (4.6)
The function S(0, p, w) is ^-periodic in w and analytic in k1.
Now, using the expansions of elliptic functions, we can obtain the expansion of the Hamil-tonian function (3.23) up to terms of order p3
K = Qp + p2Fi(6, w) + p3F6(9, w) + O(p4). (4.7)
The functions F4(9, w) and F6(9, w) depend ^-periodically on w and are analytic in the small parameter k1. The asymptotic formulae for F4(9, w) and F6(9, w) are given in the Appendix.
In what follows, we consider the nonresonant case, that is, when the inequality mQ = 2n (m = 1, 2, 3, 4; n G N) is satisfied. In this case, by a canonical change of variables p, 9 — R, ^ the Hamiltonian can be brought into the following normal form:
H = QR + c2R2 + F6($, w)R3 + O(R4), (4.8) RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2021, 17(4), 453 464_
where
2n n
C2 = ^t2j JF^e'w)dwde- (4-9)
0 0
Calculations show that, taking into account in F4(9, w) terms through order k2, we have = + (410)
The corresponding canonical transformation p, 9 — R, 1 is given by the following generating function:
U (9,R,w)= 9R + R2 U4(9,w), (4.11)
where
/ /1 x (3a + l)yfa _ y/a a(2a + 3) a
U4(9, w) = --—— sin 20 — —— sin 46»----- sin 2w--- sin 4w+
4V ' ; 24 192 16 64
a3/2(a- 2) a2 2
+ —-sin 29 cos 2w + —-- sin 2w cos 29 + O(kf). (4.12)
8(a — 1) 8(a — 1)
The function F6 in the Hamiltonian (4.8) is calculated by the formula
dF „ dU, '
F6&, w, ki) = F6 +
(4.13)
If c2 = 0, then, by the Arnold-Moser theorem [8, 9], the equilibrium point of the system (3.22) is Lyapunov stable. This implies that oscillations are orbitally stable. But if c2 = 0, then the so-called case of degeneracy takes place and solving the stability problem requires additional analysis, including terms of degree R3 in the Hamiltonian (4.8).
By solving the equation c2 = 0 for a, we have the following equation for the curve £ (see Fig. 1): 2
3 21
a* = ---ki + 0(kt). (4.14)
Now we can consider the case of degeneracy which takes place on the curve £. To this end, we put a = a^ and normalize the Hamiltonian up to terms of degree p3 by a canonical near-identity change of variables 1, R — ip, R, which is analytic in k1 and R and n-periodic in w.
The normalized Hamiltonian will take the form
H = nR + c3R3 + O(R4), (4.15)
where
Calculations show that
2n n
1
2n2
00
c3 = 77T / / F6{ip, w)dwdip. (4.16)
2363y/6 2 4
Since the quantity c3 is nonzero for sufficiently small k1, it follows that, by the Arnold-Moser theorem, the equilibrium point of the system (3.22) is Lyapunov stable. Thus, for sufficiently
small amplitudes, the pendulum-like oscillations of a rigid body are orbitally stable in the case of degeneracy.
We now turn to stability analysis of pendulum-like oscillations for arbitrary amplitude values. First, we perform the following linear change of variables:
92 = n1iQ + n12p, P2 = n2lQ + n22p,
(4.18)
where
(4.19)
nil = X12 (n), ni2 =0, n2i = A — Xii(n), %2 = Vl- A2, A = [.Tn(7r) + x22(7r)]. The functions xll(w), xi2(w), x2i (w),
x22(w) are elements of the matrizer X(w) of the linear system whose Hamiltonian is the quadratic part of (3.23).
The new Hamiltonian K*(Q, P, w) is obtained by substituting (4.18) in (3.23). In the variables Q, P the linear part of the symplectic map generated by the canonical system with the Hamiltonian K*(Q, P, w) has the simplest form. The symplectic map can be constructed in the following way. Let Q0, P0 be the initial values of the variables Q, P, and Qi, Pi be their values for w = n. Then the symplectic map generated by the canonical system with the Hamiltonian K*(Q, P, w) reads
Qi P
G
^o dP0
P + dF* + dQ0
92F4 9F4
___9F,
ap0dQ0 dP0 dp.
0
+ O6
dF, dF6 n dQl dp0 + +
6
(4.20)
where O6 denotes terms of degree six or higher, Fm = $m(Q0, P0, n), and $m(Q0, P0, w) are forms of degree m (m = 4, 6) satisfying the equalities
dw
— —G4,
=_r
dw 6 dP0 dQ0 '
(4.21)
Gm(Q0, P0, w) are forms that are obtained from Km(Q, P, w) by the change of variables
— X*(w)
Qo
Pn
(4.22)
where X* (w) is the matrizer of the linear system with the Hamiltonian K2 (Q, P, w). The forms K* (Q, P, w) are obtained by substituting (4.19) into the forms Km (m — 2, 4, 6) of the expansion of the Hamiltonian (3.23). The matrix G reads
G
cos nfi sin nfi — sin nfi cos nfi
/3 = — arccos A. n
(4.23)
Equating coefficients of the same powers on both sides of equality (4.21), we obtain twelve ordinary differential equations for 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 form and four equations for Xj(w)) in the interval [0; n], we obtain coefficients of the forms Fm. In the general case, the above system should be solved numerically. Let us introduce the following notation:
° = 3f40 + f22 + 3f04,
5
1
5
1
7 = -0/06 " 0/24 " ô/eo " 0/42 + /3i(4/o4 + 2/22 + 5/4o) + /13(4/40 + 2/22 + 5/o4)-
2
2
2
- ô [(/04 " /22 + /40)2 + (fis ~ /si)2] œt 2tTf3 - [(/13 + /31)2 + 4(/04 - /40)2] cot 7rp
(4.24)
where fj are coefficients of the form Fm (m = 4, 6). If the inequality a = 0 is satisfied, the fixed point of the map (4.20) is stable [19]. Otherwise, if a = 0, then the case of degeneracy takes place, and additional nonlinear analysis must be performed to draw conclusions on stability. In particular, it is necessary to calculate 7. If 7 = 0, then the fixed point of the map (4.20) is stable [20, 21]. Otherwise, if 7 = 0, then terms of degree eight or higher must be taken into account to answer the question of stability. As already mentioned, the case of degeneracy appears in this problem on the curve £ (see Fig. 1). Numerical calculations of the coefficients of (4.20) have shown that 7 = 0 for any values of parameters on the curve £. It yields the stability of the fixed point of the map (4.20). Thus, the pendulum-like oscillations are orbitally stable for values of parameters on the curve £. Connecting the above result with the results of a previous study performed in [11], one can formulate the following condition of orbital stability in the Bobylev-Steklov case. In the domains of linear stability, the pendulum-like oscillations of a heavy rigid body are also orbitally stable in the nonlinear sense.
5. Conclusions
The local coordinates in the neighborhood of pendulum-like periodic motions of a heavy rigid body with a fixed point can be introduced by a nonlinear canonical change of variables given by power series. It allows one to solve the problem of the orbital stability of pendulumlike periodic motions in the most difficult 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. Our study, which is based on such an approach, has shown that in the Bobylev-Steklov case, the pendulum-like oscillations that are orbitally stable in a linear approximation are also orbitally stable in a nonlinear sense.
Appendix
The expressions for F4(9, w) and F6(9, w) are
FA9, w) = - — (8cos4w + 4(2a + l)cos2w- 11)+ 16
3/2
+ ^ (6a cos2 w + 1) cos 29 - cos 49 - sin 2w sin 29 + O(kj), (5.1)
a3/2
u') = -7^(144cos8 w + 216acos6 w + (108a2 - 504a - 174) cos4 w+ 288
a3/2
+ (159a - 183) cos2 w + 214) - —— (144acos6 w + (96a2 - 240a - 32) cos4 w-
192
a3/2
- (17a - 104) cos2 w - 71) cos 29 + ^"((60«2 + 240a + 10)cos4 w~
a3/2
- (305a — 5) cos2 w — 14) cos 49 — (15a cos2 w + 1) cos 6 9+
2880
a2
+ — sin 2w(48 cos4 w + 32(a + 1) cos2 w - 89) sin 2964
a2 a2
- — sin 2w(4a cos2 w - 1) sin 49 + —— sin 2w sin 69 + O(kj). (5.2) 16 192
References
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