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5Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 4, pp. 581-594. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200404
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 34D20, 37J40, 70K30, 70K45, 37N05
On a Method of Introducing Local Coordinates in the Problem of the Orbital Stability of Planar Periodic Motions of a Rigid Body
B. S. Bardin
A method is presented of constructing a nonlinear canonical change of variables which makes it possible to introduce local coordinates in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. The problem of the orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev-Steklov case is discussed as an application. The nonlinear analysis of orbital stability is carried out including terms through degree six in the expansion of the Hamiltonian function in a neighborhood of the unperturbed periodic motion. This makes it possible to draw rigorous conclusions on orbital stability for the parameter values corresponding to degeneracy of terms of degree four in the normal form of the Hamiltonian function of equations of perturbed motion.
Keywords: rigid body, rotations, oscillations, orbital stability, Hamiltonian system, local coordinates, normal form
1. Introduction
In many problems of classical mechanics and satellite dynamics, the equations of motion admit partial solutions corresponding to plane motions. In a number of problems such motions are described by the equation of a mathematical pendulum. A well-known example of pendulumlike motions is the motion of a rigid body with one fixed point in a homogeneous gravitational
Received December 07, 2020 Accepted December 25, 2020
This work was supported by the Russian Foundation for Basic Research, project No. 20-01-00637.
Boris S. Bardin bsbardin@yandex.ru
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
field where one of its principal axes of inertia invariably occupies a horizontal position in space. Pendulum-like motions also include body motions relative to the center of mass in the central Newtonian gravitational field in a circular orbit where one of its principal central axes of inertia is perpendicular to the plane of the orbit. The above-mentioned motions have been well explored. They are either periodic motions (oscillations or rotations of the body relative to the principal axis of inertia) or motions asymptotically approaching an unstable equilibrium point. Pendulumlike periodic motions are Lyapunov unstable with respect to perturbations of the coordinate — the angle of rotation of the body relative to the axis of rotation. However, the problem of their orbital stability is of great interest from a theoretical and an applied point of view. In the above-mentioned problems the equations of rigid body motion can be represented in the form of an autonomous Hamiltonian system. Therefore, to solve the problem of orbital stability, one can apply the methods and algorithms developed to date in the theory of stability of Hamiltonian systems. In accordance with the general methods for investigating the stability of Hamiltonian systems, in a neighborhood of periodic motions it is necessary to introduce the so-called local coordinates [1] and to write equations of perturbed motion. A rigorous stability analysis of the perturbed system thus obtained can be carried out using the well-developed methods of normal forms [2, 3] and KAM theory [4, 5].
In concrete 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. One of the possible methods of introducing local coordinates is to make a canonical change of variables allowing a transformation to action-angle variables in the region of periodic motion on a two-dimensional invariant plane. If such a change of variables can be constructed in explicit form, then one can choose the perturbation of the action variable and perturbations orthogonal to the invariant plane as local coordinates describing the motion in a neighborhood of the periodic trajectory. This approach was applied, in particular, in problems of the orbital stability of pendulum-like periodic motions of a rigid body [6-12]. This approach is very efficient in cases where periodic solutions can be written in explicit analytic form. However, in using this approach one can generally encounter a number of technical difficulties, both in constructing the above-mentioned canonical change of variables and in obtaining explicit expressions for the coefficients of series expansion of a Hamiltonian in a neighborhood of unperturbed periodic motion.
Another method for introducing local coordinates was proposed in [13]. This method is used to introduce the local coordinates by making a linear canonical change of variables. It allows one to avoid the above-mentioned technical difficulties and to solve a wide range of problems of orbital stability, but in some cases its application can also be difficult due to the appearance of a singularity in the coefficients of a corresponding linear transformation. For example, this obstacle prevents this method from being applied for transformation to local coordinates in a neighborhood of pendulum-like oscillations.
In cases where equations of motion are integrated by quadratures, the orbital stability of periodic motions can be investigated using topological methods [14].
The method proposed in this paper for introducing local coordinates uses construction of a nonlinear canonical change of variables. A constructive algorithm is presented for making the above-mentioned change of variables in the form of series in powers of a new variable — a normal perturbation in the neighborhood of a periodic orbit on an invariant plane. This method makes it possible to avoid a singularity when introducing local coordinates and may be applied to investigate the orbital stability of periodic motions both in the presence of their analytic representation and in the case where periodic motions have been found numerically.
As an application of the above method, we will consider the problem of the orbital stability of periodic motions of a heavy rigid body with a fixed point in the Bobylev - Steklov case. The results of analysis of the orbital stability are in complete agreement with the results obtained previously by a different method in [10].
2. A method of transformation to local coordinates in a neighborhood of periodic motions
Consider a Hamiltonian system with two degrees of freedom dqi dH dpi dH
dt ~ dpi' dt % (i-h'2), (2.1)
where the Hamiltonian H does not explicitly depend on time.
We assume that the system (2.1) has a one-parameter family of periodic solutions of the
form
qi = f (t), Pi = g(t), q2 = P2 = 0. (2.2)
In this case the Hamiltonian function H has the following form:
H = Ho(qi,pi) + H(2) (qi pq ,P2), (2.3)
where the expansion of the function H(2) as a series in powers of the canonical variables q2, p2 begins with terms of degree at least two:
<x
H(2)(qi,Pi,q2,P2) = Y, hij(qi,Pi)q2p2 (2.4)
k=2i+j=k
We pose the problem of constructing a canonical change of variables
qi = p(e,n), Pi = ^&(e,n), (2.5)
such that in the new variables the family of periodic solutions (2.2) has the form
£ (t)= t + £(0), n = q2 = P2 = 0. (2.6)
The problem of the orbital stability of the periodic solution (2.6) 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.
We will search for the canonical change of variables (2.5) in the form of series in powers of n:
<p(£, n) = f (e) + ai(e)n + a2(e)n2 + a3(e)n3 + ..., n) = g(e)+bi(e )n+b2 (e)n2 + b3(e)n3 +....
From the condition that the change of variables (2.5) is canonical it follows that
dp d^ dp d^
(2.7)
<9£ drj drj
Equation (2.8) can be satisfied by a suitable choice of the coefficients a^Ç), b^Ç) (i = 1,2,...). For this purpose we substitute (2.7) into (2.8) and, by equating to zero the coefficients with powers of n, we obtain equations from which we can sequentially define ai(Ç), b^Ç):
oi dai 9a2 Qdg dbi db2
3&3TT7 + + 6i— - 3a3— - 2a2— - ai— = 0, (2.11)
dÇ dÇ dÇ dÇ dÇ dÇ
f n 1 ^ r\ n 1 r\j
h dJ , ^-ih dan-k dg sr i dbn-k ft
nbn kbk~ na'n~W =
From the equations thus obtained the coefficients ai(Ç), bj(Ç) are defined nonuniquely. This allows us to choose a^Ç), bi(Ç) in such a way as to avoid a singularity in the expansions (2.7). In particular, equation (2.9) can be satisfied by the following choice of a1 (Ç), b1(Ç):
dg d£
df df ai(0 = = (2.13)
where
2 ( dJf\ \ ( dg
We note that the function V2 cannot vanish. Indeed, it follows from the equation V2 = 0 that. — = — = 0. But this is impossible since the functions /(£) and g(£) define the periodic
solution to the system (2.1) and their derivatives cannot vanish simultaneously.
In a similar way one can choose a solution to Eq. (2.10). Indeed, substituting (2.13) into (2.10), we obtain
df d 2g dg d2 f
df dfdiF da di'dF2
The last equation can be satisfied by setting
d2f d2g
W2 dF2
a2(0 = 2^4, b2(0 = . (2.16)
We note that (2.13) is not a unique solution to Eq. (2.10), which allows us to avoid a singularity in the denominator.
In a similar way one can find a solution to Eq. (2.11) that contains no singularity in the denominator. This solution can be chosen, for example, in the following form:
аз(£) =
Pg д
6V(i
эе
o2f
oe
\de) dg
ш = +
&f эе
3V 8 2
+
эе
дС 2
(2.17)
df (e )
3V 8
de
The process of constructing the expansion (2.7) can be continued up to any power of n Indeed, a suitable choice of the coefficients an((), bn(() can be made, for example, as follows. Set
an(e ) = -C
dg_
bn (e) = с
c2l df
(2.18)
Now, substituting (2.18) into (2.12), we obtain an equation for finding the function C(£):
n— 1
ncv 2(e ) + kbk
dan-
n1
"n—k
k=1
de
Edbn—k
kak—^z— = 0,
k=1
de
(2.19)
whose solution yields the following expressions for an(£), bn(£):
dg
/t\ Jv^zz, 9an-k дЪп-к
= —~ S kbk~df~ ~ ^ k~
nV 2
k=1
k=1 n1
bn(o \ Y, ki"'i~jrL - Y,kbk
nV 2
k=1
k=1
de
дап-к
да
(2.20)
The right-hand sides of (2.20) are expressed in terms of ak(e), bk(e) (k = 1,2,... ,n-1) and their derivatives which were obtained at the previous steps and contain no singularities. Therefore, the expressions (2.20) contain no singularities either.
Making a change of variables by formulae (2.7), we obtain the following expansion of the Hamiltonian function in the neighborhood of n = q2 = P2 = 0:
Г = Г2 + Гз + Г4
where
r2 = n + $2(q2,P2,£), r = $3(q2,P2,e), r4 = x(e)n2 + ^2(q2,P2,e)n + $4(q2,P2 ,e). The functions x, ^ (i = 2,3,4) are given by the following explicit expressions:
x(e ) =
1
2V2
d2Hp dq2
d2Hp dp2
1
dHp dq1
i+j=2
dHp dpi dhij dH0
+ 4'
d2 Ho dHo dHo
dq1dp1 dq1 dp1 dhij dH0
dp1 dp1 dq1 dq1
ij q2 P2
(ç*,P2,e)=E hij(f(e),g(e))q2p2, k
= 2, 3, 4.
(2.21) (2.22)
(2.23)
(2.24)
(2.25)
i+j=k
2
2
The functions hij in (2.25) are the coefficients in the expansion (2.4) of the initial Hamiltonian as a series in powers of q2, p2. The partial derivatives of the functions hij and H0 in the expressions (2.23) and (2.24) are calculated for q1 = f (Ç), p1 = g(Ç).
3. Local coordinates in a neighborhood of pendulum-like motions of a heavy rigid body in the Bobylev - Steklov case
In this section we show how to apply the method described above to the problem of the orbital stability of pendulum-like motions of a heavy rigid body with one fixed point in the Bobylev - Steklov case.
Consider the motion of a rigid body about a fixed point O in a homogeneous gravitational field. Let mg be the weight of the body and let l be the distance from the center of gravity to the fixed point O. Introduce a fixed coordinate system OXYZ with the axis Z directed vertically upwards, and a moving coordinate system Oxyz, attached to the rigid body, with the axes x, y and z directed along the principal axes of inertia of the body for point O. Denote by A, B and C the corresponding moments of inertia, and by x*, y*, z*, the coordinates of the center of gravity in the moving coordinate system. Specify the position of the body (the moving axes Oxyz) relative to the fixed coordinate system OXYZ by the Euler angles 0, p.
Suppose that the Bobylev - Steklov case takes place, when A = 2C and the center of mass of the body lies on the axis x of the moving coordinate system, i.e., x* = l, y* = z* = 0.
The Bobylev-Steklov case is remarkable for the fact that the equations of motion of a rigid body admit a family of partial solutions [15, 16] which can be obtained in terms of elliptic Jacobi functions (see, e.g., [17]).
In the Bobylev-Steklov case the equations of motion also admit a partial solution describing a planar rigid body motion such that the axis of inertia z invariably occupies a horizontal position and the constant of the area integral is zero. For this motion, 0 = n/2, ^ = const, and the change in the angle p is described by the following equation of a physical pendulum:
dV . 2 n 2 mgl
-¿p+V cos^ = 0, fj, = —■ (3.1)
Thus, in this motion the body either performs pendulum-like motions about the axis z or asymptotically approaches an unstable equilibrium point.
Note that pendulum-like motions are also possible relative to the axis y. Since in the Bobylev - Steklov case the moment of inertia C is the smallest moment of inertia, we will call the motions about the axis z motions relative to the largest axis of the ellipsoid of inertia.
Choosing the Euler angles 0, p as generalized coordinates and introducing the corresponding generalized momenta p^, p$, pv, one can write the equations of motion in Hamiltonian form. In this case the angle ^ is a cyclic coordinate and the corresponding momentum p^ is a first integral and hence takes zero value for the unperturbed motion. In what follows, we assume that p^ = 0 for the perturbed motion as well.
Let us introduce a dimensionless time variable t = fit and dimensionless canonical variables qi, 52, pi, p2
Qi=<P~Y, Q2 = 0--, Pl = —, P2 = —. (3.2)
The Hamiltonian of the problem takes the form [10]
H = j [(2a — 1) sin2 qi tan2 q-2 + tan2 q-2 + 2] pf + 7(20; — 1) sin 2qi tan q-2PiP2 +
4 4 (3-3)
+ - [2a — (2a — 1) sin2 qi] — cos qi cos <72,
where a = C/B, and in the Bobylev-Steklov case | ^ a ^ 1.
For solutions corresponding to planar pendulum-like motions of the rigid body relative to the fixed axis of inertia Oz, the change of variables qi, pi is described by a system of canonical equations with the Hamiltonian H0 = 1/2p2 — cos q1, and the variables q2, p2 take zero values. Depending on the value of the constant h of the energy integral H0 = h, the planar motions are either asymptotic (h = 1) to the unstable equilibrium point p = n/2 of the rigid body or are periodic motions: pendulum-like oscillations (|h| < 1) in a neighborhood of the stable equilibrium point p = 3n/2 or rotations (h > 1) relative to the axis Oz.
Let the solution qi = f (t — T0,h), p1 = g(r — r0,h) describes the pendulum-like periodic motion of the rigid body. Then, by virtue of the system of equations with the Hamiltonian H0, the following equalities hold:
<">
Differentiating both sides of equations (3.5), we have
°2 f • f f
In a similar way we obtain
d 3f d3g
= -gcosf, = sin/(cos/ +g2). (3.6)
Taking Eqs. (3.4)-(3.6) into account, we obtain the following explicit form of the canonical change of variables (2.7):
sin f sin f 2 . ?i = / + -72^-2^ +
g g cos f 2
Pi = 9 + ytf ~-zyTfl +
sin /(cos / + g ) (g cos / + sin /) sin /
6F6 3V8
geosf (g2 cos2 / + sin2 f)g 6F6 3F8
n3 + O(n4),
(3-7)
n3 + O(n4),
where V2 = g2 + sin2 f.
The argument of the functions f and g in (3.7) is the new canonical variable £. The explicit expressions for f and g depend on the type of periodic motion. In the case of oscillations (when Ihl < 1) they are given by the formulae [8]:
f = 2arcsin[ki sn(£,ki)], g = 2ki cn(£,ki), ki = (h + 1)/2, (3.8) and in the case of rotations (when h > 1) by the formulae
f = 2am(£,k2), g = 2k2"i dn(£,k2), k2 = 2/(1+ h). (3.9) In (3.8)-(3.9), standard notation is used for the elliptic functions [18].
The period of pendulum-like motions is equal to 2n/w, where w = n/(2K(k1 )) in the case of oscillations and w = n/(k2K(k2)) in the case of rotations. K denotes the complete elliptic integral of the first kind.
We make another canonical change of variables ^ w,r by the formulae
£ = — w, rj = wr. (3.10)
In the case of rotations the Hamiltonian of the problem is a 2n-periodic function w, and in the case of oscillations it is a n-periodic function.
By substituting (3.7) and (3.10) into (3.3), we obtain the Hamiltonian of the system of equations of perturbed motion in a neighborhood of the periodic orbit
r = r2 + T4 + ... +r2m + ..., (3.11)
where r2m is a form of degree 2m in q2, p2, n1/2. The forms r2 and r4 of the expansion (3.11), which are required for further analysis, have the form
r2 = wr + $2(q2 ,P2,w), (3.12)
r4 = w2x(w)r2 + w^2(q2,P2,w)r + $4(q2,P2,w), (3.13)
where
*2(q*,P2,w) = Yl ^ijq2p2, (q2,P2,w)= Y <Pijq2P2, k = 2,4. (3.14)
i+j=k i+j=k
The coefficients of the forms $2, $4 which periodically depend on w are calculated by the formulae
= éi{cos f* ~ ^sin2 ^ ~ 9 02o(«O = t(sin2 f*(cos f + X)(2a " !) + ~ sin2 /*] ' 0n(w) = -—^ (2a - 1 )g* sin / [sin2 /* - cos /(cos /* + 1)], Ip02(w) = (2a - 1) sin2 /* cos /*, <P2o(w) = i [(2a - (2a - 1) cos2 /*)#2 + 2 cos /*], <pn(w) = ^(2a - 1 )g* sin/* cos/*, <fio2(w) = j [(2a - l)cos2/* + l],
(3.15)
<P4o(w) = ^
(2a sin2 /* + cos2 /*)#* - j cos /*
<fi3i{w) = ^(2a - 1 )g* sin/* cos/*, p22(w) = LpVi(w) = <Am(w) = 0, 6
where f*, g* denotes the functions f*(w) = f(w-1w), g*(w) = g(w-1 w), T-periodic in w, with T = 2n in the case of rotations and T = n in the case of oscillations.
By virtue of the equations of motion with the Hamiltonian (3.11), the coordinate w is an increasing function of the variable t. Therefore, in the problem of the stability of motion this coordinate can play the role of time. 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 + ..., (3.16)
where Km is a form of degree m in q2, p2 with coefficients T-periodic in w. The forms K2 and K4 have the following explicit form:
K2 = -$2(q2,p2,w), (3.17)
u
K4 = - [x(w)<S>22(q2, p2,w) + ^2(q2, p2,w)<S>2(q2, p2,w) + <$>i{q2, p2, w)] • (3.18) u
The equations of motion on the isoenergetic level r = 0 can be written in the Hamiltonian
form
dq2 dK dp2 dK
dw dp2' dw dq2 (3.19)
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 (3.19).
4. On the orbital stability of pendulum-like periodic motions of a heavy rigid body in the Bobylev - Steklov case
The problem of the stability of pendulum-like periodic motions of a heavy rigid body in the Bobylev - Steklov case has been investigated previously. In [9], the case of pendulum-like motions relative to the axis Oy was discussed. The case considered here of pendulum-like motions relative to the axis Oz was examined in [10].
In the above-mentioned papers, the local coordinates were introduced by first making a canonical change of variables which allows introduction of action-angle variables in the region of phase space corresponding to unperturbed periodic orbits. 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 normal perturbations to the invariant manifold on which periodic orbits lie. In such an approach, it is necessary to have an explicit analytic representation of the unperturbed periodic motion. This is mainly due to the fact that 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 degenerate cases 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 that reason, in particular, the above-mentioned degenerate case was not considered in [9, 10].
In [10], a nonlinear analysis of the orbital stability of the pendulum-like motions considered here was carried out taking into account terms through order four in the expansion of the Hamiltonian in a neighborhood of the periodic orbit. We give a brief account of the main results of [10]. 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 r, on which a degeneracy takes place. For parameter values on this curve, the problem of orbital
a
Fig. 1. Stability diagram for pendulum-like motions of a rigid body in the Bobylev - Steklov case.
stability is solved using terms of order six, or perhaps of higher orders, in the expansion of the Hamiltonian of the perturbed motion.
The boundaries separating the stability and instability regions were obtained in the following analytic form.
• The left and right boundaries of the region of orbital instability of pendulum rotations are given, respectively, by the equations
h = — -—-—-, h =
2a- 1 2a- 1
• The regions of orbital stability and instability of pendulum-like oscillations are separated by the straight line h = 0.
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 r. In this paper we consider the case of small oscillation amplitudes, when it is possible to introduce a small parameter and perform the analysis of orbital stability analytically.
As a small parameter of the problem we choose the quantity k\ = sin /3/2 (where /3 is the amplitude of oscillations). This quantity is the absolute value of the elliptic functions in the expressions (3.8) and is related to the constant energy h by kf = (h + 1)/2. In order to investigate the stability of the equilibrium point in the degenerate case, it is necessary to reduce the Hamiltonian (3.16) 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 [4, 5].
At the first stage of the normalization procedure we present the quadratic part of the Hamiltonian (3.16) to the form corresponding to a harmonic oscillator. Using the well-known
expansions of elliptic functions [18], we obtain from (3.17)
2
By making the change of variables
K'2 = \ql + -ap2 + 0(k2). (4.1)
I _l
q-2 = сиж, P2 = а 4У (4.2)
we reduce the autonomous part K2 to the required form:
K2 = + У2) + 0{k\). (4.3)
By making a canonical near-identity linear ^-periodic in w change of variables x,y ^ X,Y
x =an(w)X + a12(w)Y
(4.4)
y =a2i(w)X + a22(w)Y
we can exclude the dependence of K2 on w and bring the quadratic part of the Hamiltonian into the following form
К2 = ^П{Х2 + Y2), (4.5)
The functions a11 (w), a12 (w), a21(w), a22 (w) are ^-periodic in w and analytic in k1. They can be constructed in form of converging series in powers of the small parameter k1. The coefficients of any finite power of k1 in these series can be determined, for example, by the Birkhoff method [1, 3] or the Deprit-Hori method [2]. The calculations have shown
. . 1 — 3a + 2a2 + a (3 — 4a)cos2w, 2 fflll(№) =1 +-4^=1)-+
. . Ja (7 - 8a) sin2w , 2
= 4(a-i)-+
. . (2 - 3 a)sin2w, о
a2l(W) = ^L__k2 + 0(ki),
(4.6)
. . 1 — 3a + 2a2 + a (3 — 4a)cos2w, 2
°22 = 1--4^=1)-+
and
П = (4.7)
We now transform to canonical polar coordinates p by the formulae
X = \[2p sin Y = л/2р cos :d (4.8)
and, using the expansions of elliptic functions, obtain the following explicit form of the expansion of the Hamiltonian function through terms p3:
K = Qp + p2 F4 (tf,w) + p3 F6 (tf,w) + O(p4), (4.9)
where
Fi{9, w) = --acos4 ft — y/a (2 a- 1) sin2w cos ft sin ft + (sin2 w — 4/3a) sin2 ft +
236 5 1 (4.10)
i 23 -2 5 2 1 4 , r\ti2\
+ — a — sin w — —acos w —-acos w + (J(k1),
, 1152a3 + 677a2 + 1392a + 324 w . n . .
F6(0,w) =--+ M6(0,w) + O(k21). (4.11)
By M6(ft,w) we denote trigonometric terms, whose mean values over periods of variables ft, w are equal to zero. These terms are not used further.
Since the Hamiltonian (4.9) depends ^-periodically on w, it follows that, if the equation
mQ = 2n, m,n e N, (4.12)
is satisfied, resonance takes place in the corresponding canonical system. In the case of resonance, the normal form of the Hamiltonian contains additional (resonant) terms; therefore, resonance cases are considered separately.
In normalizing the Hamiltonian through terms of degree p2 it is necessary to take into account only resonances of orders not higher than four (m ^ 4). Higher-order resonances do not manifest themselves in this approximation. The Hamiltonian function (4.9) contains no terms of order p3/2, which implies that its normal form calculated through terms of order p2 will also contain no terms corresponding to a resonance of order three, 3Q = 2n. We finally note that, when ki ^ 1, first- and second-order resonances, corresponding to boundaries of the regions of parametric resonance, and a fourth-order resonance are impossible since the inequality 1/3 ^ a < 1 is satisfied. Thus, in the case of small-amplitude oscillations with any admissible values of the parameter a, the Hamiltonian (4.9) can be reduced to the following normal form by a suitable choice of canonical variables:
H = QR + C2R2 + F6(^,w)R3, (4.13)
where
2n n
1
C'2
J J F4(ft,w) dwdft. (4.14)
2n2
0 0
The change of variables ft,p — ^, R, which reduces the Hamiltonian (4.9) to the form (4.13), is a near-identity transformation analytic in k1 and p, n-periodic in w and 2n-periodic in
Calculations show that by taking into account in F4(ft,w) terms up to order k2 inclusively we have
1 3 (45 a3 - 29 a2 — 16 a + 12) 2 , 4n ,
The corresponding canonical transformation ft, p — is given by the following generating function:
5(ft, R) = ftR - R2S4(ft,w), (4.16)
where
x >/a o (24a (4a - 3) cos2 w - 77a2 + 77a - 12) ,
S4(ft,w) =^sinftcos3ft + ^---48-y/a (a—1)-^ sin ft cos ft + (4.17)
sin 2w (4a2 — 2a — 1) sin2 ft sin 2w (2a (a — 1) cos2 w — 3a2 — 9a + 8) 2
4(^1) + 32(a - 1) + °{kl
The function F6 in the Hamiltonian (4.13) is calculated from the formula
F6(^,w,k! ) = F6 +
0F4 0S4
(4.18)
If c2 = 0, then by the Arnold-Moser theorem [4, 5], the equilibrium point of the system (3.19) 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 problem of the stability requires an additional analysis including terms of degree p3 in the Hamiltonian (4.13).
By solving the equation c2 = 0 with respect to a we have the following equation for curve r (see Fig. 1)
2 k2
<x* = T + 1t + 0(kt). (4.19)
36
Let us now consider the case of degeneracy which takes place on curve r. To this end we put a = a*. Then by substituting (4.10), (4.11), (4.17) into (4.18), one can obtain an explicit expression for F6 as series in powers of k1. With an explicit expression for F6 in place, we can normalize the Hamiltonian up to terms of degree p3 by a canonical near-identity change of variables ^,R — ip,R,, analytic in k1 and R and n-periodic in w.
The normalized Hamiltonian will take the form
H = QR + C3R3 + O(R3), (4.20)
where
2n n
°3 = ¿2 / (4.21)
00
Calculations show that
= + (4-22)
384 9216
Since the quantity c3 is nonzero for sufficiently small ki, it follows that, by the Arnold-Moser theorem, the equilibrium point of the system (3.19) is Lyapunov stable and hence the corresponding pendulum-like oscillations are orbitally stable as well.
Thus, in the Bobylev- Steklov case, pendulum-like periodic oscillations of a rigid body relative to the largest axis of an ellipsoid of inertia with sufficiently small amplitudes are orbitally stable.
References
[1] Birkhoff, G. D., Dynamical Systems, Providence, RI: AMS, 1966.
[2] Giacaglia, G. E. O., Perturbation Methoda in Non-Linear Systems, Appl. Math. Sci., vol.8, New York: Springer, 1972.
[3] Markeev, A. P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).
[4] Siegel, C. and Moser, J., Lectures on Celestial Mechanics, Grundlehren Math. Wiss., vol. 187, New York: Springer, 1971.
[5] 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.
[6] Markeev, A. P., Stability of Plane Oscillations and Rotations of a Satellite in a Circular Orbit, Cosmic Research, 1975, vol. 13, no. 3, pp. 285-298; see also: Kosmicheskie Issledovaniya, 1975, vol. 13, no. 3, pp.322-336.
[7] Markeev, A. P. and Bardin, B. S., On the Stability of Planar Oscillations and Rotations of a Satellite in a Circular Orbit, Celest. Mech. Dynam. Astronom., 2003, vol.85, no. 1, pp. 51-66.
[8] Markeyev, A. P., The Stability of the Plane Motions of a Rigid Body in the Kovalevskaya Case, J. Appl. Math. Mech., 2001, vol.65, no. 1, pp.47-54; see also: Prikl. Mat. Mekh, 2001, vol.65, no. 1, pp. 51-58.
[9] Bardin, B. S., Rudenko, T. V., and Savin, A. A., On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev- Steklov Case, Regul. Cahotic Dyn., 2012, vol. 17, no. 6, pp. 533-546.
[10] Bardin, B. S., On the Orbital Stability of Pendulum-Like Motions of a Rigid Body in the Bobylev -Steklov Case, Regul. Chaotic Dyn., 2010, vol. 15, no. 6, pp. 702-714.
[11] Bardin, B.S. and Savin, A.A., On the Orbital Stability of Pendulum-Like Oscillations and Rotations of a Symmetric Rigid Body with a Fixed Point, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3-4, pp. 243-257.
[12] Bardin, B. S. and Savin, A. A., The Stability of the Plane Periodic Motions of a Symmetrical Rigid Body with a Fixed Point, J. Appl. Math. Mech., 2013, vol. 77, no. 6, pp. 578-587; see also: Prikl. Mat. Mekh., 2013, vol.77, no.6, pp. 806-821.
[13] Markeyev, A. P., An Algorithm for Normalizing Hamiltonian Systems in the Problem of the Orbital Stability of Periodic Motions, J. Appl. Math. Mech., 2002, vol. 66, no. 6, pp. 889-896; see also: Prikl. Mat. Mekh., 2002, vol. 66, no. 6, pp. 929-938.
[14] Bolsinov, A. V., Borisov, A.V., and Mamaev, I.S., Topology and Stability of Integrable Systems, Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259-318; see also: Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71-132.
[15] Bobylev, D.N., On a Particular Solution of the Differential Equations of a Heavy Rigid Body Rotation around of a Fixed Point, Tr. Otdel. Fiz. Nauk Obsch. Lyubit. Estestvozn., 1896, vol. 8, no. 2, pp. 21-25 (Russian).
[16] Steklov, V. A., A Certain Case of Motion of Heavy Rigid Body Having Fixed Point, Tr. Otdel. Fiz. Nauk Obsch. Lubit. Estestvozn., 1896, vol.8, no. 1, pp. 19-21 (Russian).
[17] Borisov, A.V. and Mamaev, I.S., Rigid Body Dynamics, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2001 (Russian).
[18] Gradshtein, I.S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 7th ed., Amsterdam: Acad. Press, 2007.
