ON NONLINEAR RESONANT OSCILLATIONS OF A RIGID BODY GENERATED BY ITS CONICAL PRECESSION Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 4, pp. 503-518. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd180406

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 70H09, 70H12, 70H14

On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession

A. P. Markeev

The motion of a dynamically symmetric rigid body relative to its center of mass in the central Newtonian gravitational field in a circular orbit is investigated. This problem involves motion (called conical precession) where the dynamical symmetry axis of the body is located all the time in the plane perpendicular to the velocity vector of the center of mass of the body and makes a constant angle with the direction of the radius vector of the center of mass relative to the attracting center. This paper deals with a special case in which this angle is n/4 and the ratio between the polar and the equatorial principal central moments of inertia of the body is equal to the number 2/3 or is close to it. In this case, the conical precession is stable with respect to the angles that define the position of the symmetry axis in an orbital coordinate system and with respect to the time derivatives of these angles, and the frequencies of small (linear) oscillations of the symmetry axis are equal or close to each other (that is, the 1:1 resonance takes place).

Using classical perturbation theory and modern numerical and analytical methods of nonlinear dynamics, a solution is presented to the problem of the existence, bifurcations and stability of periodic motions of the symmetry axis of a body which are generated from its relative (in the orbital coordinate system) equilibrium corresponding to conical precession. The problem of the existence of conditionally periodic motions is also considered.

Keywords: resonance, stability, oscillations, canonical transformations

Received July 26, 2018 Accepted August 25, 2018

This work was carried out within the framework of the State Assignment (registration No. AAAA-A17-117021310382-5) and was partially supported by the Russian Foundation for Basic Research (project No. 17-01-00123).

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 shosse 4, Moscow, 125080 Russia

Moscow Institute of Physics and Technology (State University)

Institutskii per. 9, Dolgoprudnyi, Moscow region, 141700 Russia

1. Introduction

A rigid body moving in the central Newtonian gravitational field is considered. The characteristic linear dimensions of the body are assumed to be small in comparison with the distance from the center of mass of the body, O, to the attracting center O*. This allows the assumption [1] that the rotational motion of the body relative to the center of mass does not influence the motion of the center of mass. We assume that the orbit of the center of mass is circular, and denote the angular velocity of motion of the center of mass in the orbit by u0.

Let OXYZ be a trihedron of an orbital coordinate system with the axis OZ directed along the radius vector O*O of the center of mass of the body and with the axes OX and OY directed along the velocity vector of the center of mass and along the normal to the plane of the orbit, respectively. Let us introduce another trihedron, Oxyz, with axes directed along the principal central axes of inertia of the body, and let A, B and C be the moments of inertia of the body relative to the axes Ox, Oy and Oz. The relative orientation of the trihedrons OXYZ and Oxyz will be given by the Euler angles p, 9, p introduced in a standard way.

Assume that the body is dynamically symmetric (A = B). Then the angle of proper rotation, p, will be a cyclic coordinate and, as a consequence, the projection r of the absolute angular velocity of the body onto the axis of its dynamical symmetry is constant throughout the motion:

r = tp cos 9 + p — u0 cos p sin 9 = ro = const. (1.1)

(The dot denotes differentiation with respect to time t.)

To describe the rotation of the body relative to its center of mass, we will use the Hamil-tonian form of the equations of motion. If we take v = u0t as an independent variable and nondimensionalize the momenta canonically conjugate to p, 9 using the multiplier Au0, then the Hamiltonian function can be written as [2, 3]:

H = \ —%-n + \ Pe + M^r?; + cos i> cot d) 'P4> - sin + 2 sin2 9 2 sin2 9

+ J a2/32 cot2 9 + a/3^4 + ha-l) cos2 9, ^

2 sin 9 2

a = J> ¡3 = TJ-0 (0 < a ^ 2, -oo < ¡3 < oo).

The equations given by the Hamiltonian function (1.2) describe the motion of the symmetry axis of the body relative to the orbital coordinate system. If this motion is found, then the rotation of the body about the symmetry axis is defined from (1.1) using a quadrature.

There exist (see, e.g., [2, 3]) three types of solutions corresponding to the motion of the body in which its symmetry axis is fixed in the orbital coordinate system (p = p0 = const, 9 = 90 = const). For these motions, the symmetry axis: 1) is either always perpendicular to the plane of the orbit, 2) or lies in the plane perpendicular to the radius vector of the center of mass, 3) or lies in the plane perpendicular to the velocity vector of the center of mass of the body. Motions 1), 2) and 3) correspond to regular (respectively, cylindrical, hyperboloidal and conical) precessions of the body in absolute space; the axis of precession is the axis passing through the center of attraction and perpendicular to the plane of the orbit, the angular velocity of precession coincides with the angular velocity u0 of motion of the center of mass in the orbit, and the angular velocity of proper rotation is defined by the equation p = (0 + cosp0 sin 90)u0.

In motions 1), 2) and 3), the symmetry axis of the rigid body stays in absolute space on the surface of a cylinder, a one-sheeted hyperboloid and a cone, respectively.

The problem of stability of the above-mentioned precessions with respect to perturbations of the quantities p, d, ip, d has been examined in detail. Relevant references can be found in [2, 3].

This paper investigates nonlinear resonant oscillations of the symmetry axis of a body which are generated from its equilibrium point in the orbital coordinate system which corresponds to the conical precession of the body. This precession corresponds to the solution of a system of equations with the Hamiltonian function (1.2) in which

aft = (3a — 4) sin d0, p0 = 0, pe = pec =0, p^ = p^0 = 3(a — 1) sin d0 cos d0.

For this solution, the symmetry axis of the body lies throughout the motion in the plane OYZ of the orbital coordinate system and makes a constant angle d0 with the direction of the radius vector of the center of mass of the body.

If the rigid body is dynamically elongated along its symmetry axis Oz, that is, if the inequality a < 1 is satisfied, then the Hamiltonian function (1.2) is definite positive in a neighborhood of the conical precession and stability takes place. Inside the stability region {a < 1, 0 < d0 < n/2} the frequencies w1, w2 of small oscillations of the symmetry axis are defined by the equation

w4 — [7 — 6a + 9a(a — 1) sin2 do]w2 + 3(a — 1)(3a — 4) cos2 do = 0.

The quantities w1 and w2 do not vanish and are different. An exception is one point, a = 2/3, d0 = n/4, at which the frequencies w1 and w2 are equal to each other (that is, at this point the 1:1 resonance takes place).

In this paper it is assumed that the quantity a is close or equal to its resonant value 2/3 and the problem of the existence, bifurcations and stability of periodic motions of the symmetry axis is investigated. Some problems concerning conditionally periodic motions are also discussed. The study uses classical perturbation theory, the Deprit-Hori method as well as other numerical and analytical methods and algorithms of nonlinear dynamical systems.

The problem of nonlinear oscillations of Hamiltonian systems in the presence of resonance has a more than 100-year history [4-6]. Relevant references to classical and recent works that contain general theoretical conclusions and their applications to specific problems in mechanics can be found, for example, in [7]. Among the studies on the 1:1 resonance in stable Hamiltonian systems, we mention the papers [8-11] dealing with several model problems important for astrophysics and celestial mechanics. The 1:1 resonance in the problem of motion of a rigid body near its hyperboloidal and cylindrical precessions in a circular orbit has recently been explored in [7, 12].

2. The Hamiltonian function of perturbed motion and its normal form

Set

d = do + e£i, p = , Pe = £Vi, P^ = P^c + £V2 (0 <e < 1), (2.1)

/3 = do + sin do(3a — 4)/a, do = n/4, a = 2/3 +e2d. (2.2)

The quantity e2d is a deviation of the dimensionless inertial parameter a from its resonant value, which is equal to 2/3. The parameter / in (2.1) is such that for any value of d the rigid body can perform a conical precession in which 90 = n/4.

Equations (2.1) define a canonical transformation with the valence e-2. From (1.2) and (2.1), (2.2) we find that the new canonically conjugate variables (i, ni (i = 1) 2) correspond to a Hamil-tonian function that can be represented as a power series in the parameter e:

H = \&- + - bvi + \ril + 4 +1-( - + Him - - + lêm) +

+ e

+ e3

f d( - + 4m - el) + f £ - f tim + + - telm - +

- 2^Г]2) ~ -g-Çi + -J2 Çil'2 - - +

+ (2.3)

+ + - ¿fin] + o{é).

The canonical univalent change of variables

6 = -^1 + ^2, b = + V1 = V3U2, m = -^Ul (2.4)

reduces the Hamiltonian function (2.3) to the form

G = G(0) + eG(1) + e2G(2) + e3G(3) + O(e4),

(2.5)

where G(0) is a real normal form of the function (2.3) with e = 0 which is the sum of Hamiltonians of two identical harmonic oscillators,

G{0) = l(it + U2) + l(v2 + U2),

and G(k) (k = 1,2,3) in the expansion are functions that are coefficients with ek in the expansion (2.3) in which the quantities ni (i = 1,2) have been replaced with their expressions (2.4).

The following transformation ui,Ui — qi,pi (i = 1,2) performs normalization [13] of the Hamiltonian function (2.5) in terms up to the fifth degree in ui, Ui. This transformation eliminates first- and third-degree terms in e and simplifies second-degree terms. The change of variables ui,Ui — qi,pi was constructed by the Deprit-Hori method [14] modulo third-degree terms in e. We write out this change of variables, omitting for brevity terms higher than the first degree:

ui =qi- ejg(2qi + 'Ml + 3pj - 2^2pip2 + 2p2),

U2 = 92 - (8<?i<72 - 7V2q2 - 2V2p2 + 8 pm - 18V2p2),

Ui = pi + e-g-(qiPi + q2P2), U2 = P2 + P2 + Ч2Р1 ~ 7\/2<?2j92)-

In the symplectic polar coordinates Ti given by the equations

qi = v/2rlsin Lpi, pi = v/2n cos ipi (i = 1,2),

(2.6)

(2.7)

(2.8)

(2.9)

the normalized Hamiltonian function (2.3) can be written as

H = n + r2 + £2

|dr2 - ^r'l - yrir2 - j|r| +

(2.10)

+ "Y^ri/2r2/2(8ri + 15r2)cos((^i - If 2) - ^rir2cos2((^i - <£2) + 0(t4)-

3. Simplification of the normal form

If we neglect the last term in (2.10), we arrive at an approximate system, which, as is easy to check, possesses the integral r1 + r2 = const. We make use of this fact in order to simplify the normal form (2.10). To do so, we perform a canonical univalent transformation $i,ri — Xi,Ri which is given by the equations

Xi = fi — $2, X2 = $2, Ri = ri, R2 = ri + r2. (3.1)

In the variables Xi, Ri the Hamiltonian function (2.10) can be written as

H = Fi + F2 + O(e4), (3.2)

where

Fi = -|i[8Ei(E2 - Ri) cos2 xi - 2y/2r\/2{R2 ~ Ei)1/2(15E2 - 7Ri) cos Xi ~ (3 3) — 8R2 — 12Ri R2 — 81dRi ],

F2 = R2 - |g(81 d + 26R2)R2. (3.4)

In an approximate system with a Hamiltonian function obtained by neglecting the quantities O(e4) in the function (3.2), the coordinate x2 is cyclic and there exists the integral R2 = c> 0 — const.

4. On an auxiliary (model) system with one degree of freedom

In the above-mentioned approximate system the variables Xi, Ri correspond to a one-degree-of-freedom system with canonical equations of motion given by the function Fi from (3.3), in which R2 = c. If we introduce new variables x, r instead of Xi, Ri by the formulae

Xi = X, Ri = cr (0 ^ r < 1) (4.1)

and take the quantity t = (s2c/36)v as an independent variable, then the Hamiltonian function of the model system takes the form

T = —8r(l - r) cos2 x + 2\/2 v/r(l -r)(15 - 7r) cos x + 8r2 + 12r + 815r, (4.2)

where we have introduced the notation d = cS. In the variables q,p defined by the equations

q = V2rsmx, p = V2reosx, (4.3)

we have the following expression for the function (4.2):

T = |(4+27i)g2+i(4+81i)p2+2(g4+3gV+2/)+^(30-7g2-7p2) V2 - q2-p2. (4.4)

The equations of motion of the model system have the first integral

r = h = const. (4.5)

On equilibrium points of the model system. The equations of motion of the model system which are given by the Hamiltonian function (4.2) (or (4.4)) admit partial solutions — the equilibrium points % = xo, r = r0 (or q = q0, p = p0). These solutions are defined by the equations

cT n dT n (nr dT n dT ,A rs

■7T- = 0, T— = o or — = 0, — = o . (4.6)

or ox \ dq dp J

It follows from (4.6) that the quantity r cannot be zero at equilibrium points and that the second equation of (4.6) can be written as

v/2(15 - 7r)

cos x

— i

8s/r(l-r)

sin x = 0. (4.7)

The minimal value of the first term in the square brackets is equal to \/l5 (it is attained at r = 15/23). Therefore, Eq. (4.7) is equivalent to the equation sinx = 0 and there exist only two types of equilibrium values Xo = X^ (k = 1,2) that are physically different from one another:

x01) =0 and x02) = (4.8)

Note that the dependence between the equilibrium value r0 and the quantity S = Sk is found from the first equation of the system (4.6), in which we need to set x = Xo^ (k = 1,2). Thus, we obtain

fc ^ 4(ro) = -¿(1 + 8,0) + (-D* ^(28r°2-;51r°V5) (k = 1,2). (4.9)

81 81v/r0(l - r0)

The graphs of functions S1 and S2 are shown in Fig. 1. The graph of function S1 intersects the abscissa axis at the point r0 = 0.835, and the graph of function S2 intersects it at the point r0 = 0.256. Each of the graphs has two vertical asymptotes: r0 = 0 and r0 = 1.

At the equilibrium point of the kth type the constant h of the integral (4.5) is defined by

h, = Mn») = -l«i + (-1)* --B (t = l,2). (4.10)

\/1 — ro

For equilibrium points of both types, q0 = 0 and the quantity S is expressed in terms of the equilibrium value p0 by the formula

i = = (4,1)

81 81p0\/2 - p2

Equation (4.11) is presented graphically in Fig. 2. The graph intersects the ordinate axis at the points po = 1.292 and po = —0.716 and has three horizontal asymptotes: po = 0, po = \/2 and Po = -\/2.

As is seen from Figs. 1 and 2, there exist two equilibrium points for any value of 5: at the

first, of them, xo = 0, r = ro (qo = 0, po = V^ro), and at the second, xo = ir, r = tq (qo = 0, Po = -V^ro).

Analysis has shown that for equilibria of both types the roots of the characteristic equation (linearized in a neighborhood of the equilibrium equations of motion of the model system) are purely imaginary. Therefore, the equilibria found in the phase plane correspond to singular points of center type.

5. On the set of admissible values of the quantities 5 and h in the integral (4.5)

A straightforward analysis based on the fact that | cos x\ ^ 1 and 0 ^ r(1 — r) ^ 1/4 shows that the set of admissible values of 5 and h consists (see Fig. 3) of two regions g\ and g2 and their boundaries I and 71, y2. The boundary curves Yk (k = 1) 2) are given parametrically using the equations 5 = 5k(r0), h = hk(r0), where 0 < r0 < 1, and functions 5k and hk are defined

i-1---1-1-1

ft fl /1 On flO n A ng

Fig. 3. Regions of admissible values of S and h in the integral (4.5).

by Eqs. (4.9) and (4.10). The boundary t shown in Fig. 3 is the straight line h = 20 + 81S, which separates regions g1 and g2. It is the common asymptote of the curves 71 and 72; the abscissa axis h = 0 is also the common asymptote of these curves. Region gk is a collection of points lying between the asymptote t and the curve Yk (k = 1,2).

For values of S and h lying to the left and above the curve y1 or to the right and below the curve y2, no motion is possible.

The points of the curve 71 correspond to equilibria of the first type: qo = 0, po = \/2ro, and the points of the curve 72 correspond to equilibria of the second type: qo = 0, po = — \/2ro-If the values of S and h lie in region gk (k = 1,2), then in the phase plane the trajectories of the model system are closed curves which encircle the equilibrium point of the kth type. These two sets of curves are divided by a curve — a separatrix given by the equation

The separatrix connects the points q = — \/2, p = 0 and q = \/2, p = 0.

When the parameters S and h cross the straight line t, the phase portrait undergoes a bifurcation. If they cross it, for example, from region g2 to region gi , then a family of phase curves encircling the singular point q = 0, p = — \/2ro disappears and a family of curves encircling the point q = 0, p = \/2ro appears.

Figure 4 shows phase portraits for several values of the parameter S. The phase trajectories are located inside the circles q2 + p2 = 2 shown in Fig. 4 by dotted lines. The dashed lines indicate separatrices (5.1).

6. Families of periodic motions in a complete system

The equilibrium points X = X0, r = r0 of the model system correspond to the following solutions of the approximate system with the Hamiltonian function F1 + F2 (see Section 3):

V2p (30 - 7q2 - 7p2) - [4{q2 + 2p2) + h] \J'2 - q2 - p2 = 0.

(5.1)

Ri = cro, R2 = c, Xi = x0k), X2 = ^(k) (k = 1,2).

(6.1)

20 81

Fig. 4. Phase portraits of the model system.

¿ = 0

Here x0k) is given by (4.8), and

a(k) = u(k) v + X20 (X20 = const),

where

(k) = 1 _

S2C

36y/rS

[12^(4 - 3r0) + (-l)fc 3v/2(l-r0)(5 + 3r0)].

(6.2)

(6.3)

Using the Poincare method, as in [15, 16], it can be shown that in a complete system with the Hamiltonian function (3.2) there exist two solutions analytic in e and tending, as e — 0, to the solutions (6.1). As in the cases of equilibrium of the model system, we shall call them periodic motions of the first type (if in (6.1) k = 1) or of the second type (if k = 2). These families of motions depend on the parameter c and, when c = 0, degenerate to the equilibrium 0o = n/4, Po = 0 (see (2.2)).

It follows from (6.1) and the change of variables (2.4), (2.6)-(2.9), (3.1) that on the periodic motion of the kth type the quantities £i, £2, which characterize (see (2.2), (2.3)) the deviation of the symmetry axis of the body from its equilibrium point d0 = n/4, p0 = 0 in the orbital coordinate system, are given by the following expressions:

6 = ^ + (-1)' yßrÖ\ Sin <7* +

+ || [11 - 7r0 + (-l)fc 4v/2ro(l-ro)] (3 + cos 2ak) + 0(e2),

6 = ^ [v/2(l - r0) - (-l)fc y/rt] cos ^ -

3

2ec 9

(6.4)

(6.5)

" ^F [5 " 7r° + 4V2ro(l - r0)] sin 2ak + 0(e2).

On trajectories of periodic motions in the plane £2, For a qualitative analysis of the trajectories, we neglect in (6.4) and (6.5) the first- and higher-order terms in e. Then the approximate expressions for the abscissa £2 and the ordinate of the point M of the trajectory of periodic motion of the kth type will be given by the first terms of (6.5) and (6.4).

As we change the quantity r0 (or, which is the same, the quantity 5 related to r0 by (4.9)), the shape of the trajectory of periodic motion in the plane £2, changes.

1. We first consider periodic motions of the first type. In the interval 0 <r0 < 1/3 < —2/9) the trajectory is an ellipse with semiaxes k2 and k1 which are calculated by the formulae

K2

2 a/3c

V6c,

3

[y/2(l-r0) + y/¥E], = ~ V2Fol

(6.6)

The ellipse is elongated along the abscissa axis, and point M moves along the ellipse counterclockwise (Fig. 5a). With increasing ro in the above-mentioned interval the semiaxis k2 increases from 2\/6c/3 to 2-y/c, and the semiaxis k,\ decreases from \/6c/3 to zero.

(a)

(b)

(c)

a

Kx

c

ii «2

0<ro<| r0 = | r0>|

Fig. 5. Periodic motions of the first type in the plane £2, £1.

If ro = 1/3 (S = —2/9), then the trajectory is a segment of the axis £2 of length 4yjc. (Fig. 5b). As r0 passes through the value 1/3, the motion of point M along the trajectory reverses direction.

When r0 > 1/3 (5 > —2/9), the trajectory is again an ellipse elongated along the axis £2, point M moves along the ellipse clockwise (Fig. 5c). With increasing r0 the semiaxis k2 decreases, and the semiaxis k\ increases; in the limit as ro —> 1 the trajectory of point M tends to a circle of radius 2\/3c/3.

2. For motions of the second type the evolution of the shape of trajectories in the plane £2, £1 is somewhat more diverse than in the case of motions of the first type. But as before, the trajectories are, as a rule, ellipses, with the difference that for several values of r0 they become circles or degenerate to a straight-line segment. The semiaxes of the ellipses are calculated by the formulae

«2 = ^lv/2(l-r0)-^|, = + (6.7)

In the interval 0 < r0 < 1/9 (5 > 4/9) the ellipse is elongated (as in the case of motions of the first type) along the axis £2. With increasing ro in this interval k2 decreases from 2\/6c/3 to 2\/3c/3, and m increases from \/6c./3 to 2\/3c/3. Point M moves along the ellipse counterclockwise (Fig. 6a). When ro = 1/9 (S = 4/9), the ellipse becomes a circle of radius 2\/3c/3.

In the interval 1/9 < r0 < 2/3 (—5/9 < 5 < 4/9) the ellipse is elongated along the axis £1, and point M moves, as before, counterclockwise (Fig. 6b). With increasing r0 the semiaxis k2 decreases from 2\/3c/3 to zero, and k\ increases from 2\/3c/3 to \/2c. When ro = 2/3 (S = —5/9), the ellipse degenerates to a segment of the axis £1 of length 2\/2c (Fig. 6c).

(a)

(b)

(c)

(d)

K1 r2

J K 2

0<ro<±

9<ro<3

—vie

r0 = 3

r-0>3

Fig. 6. Periodic motions of the second type in the plane £2, £1.

As r0 passes through the value 2/3, the motion of point M along the trajectory reverses direction. When ro > 2/3 < —5/9), the ellipse is elongated along the axis £1, and point M moves clockwise (Fig. 6d). With increasing r0 the semiaxis k2 increases and k1 decreases and, as a result, as ro —> 1, the trajectory tends to become a circle of radius 2\/3c/3.

On the stability of periodic motions. To analyze the orbital stability of periodic motions, we use the Arnold-Moser theorem [17] on the stability of the Hamiltonian system with two degrees of freedom. A constructive algorithm for such an analysis is set forth in detail in [18, 19]. We omit the details of calculations and describe only the results obtained therein.

For the stability analysis of periodic motions of the kth type, the quantity given by (6.2) and (6.3) was taken as an independent variable, and by a suitable choice of real canonically conjugate variables Qj, fij (j = 1,2) the Hamiltonian function of perturbed motion was brought to its normal form

$(Qi,Q2,$1,$2) = Qi + AQ2 + C20Q2 + C11Q1Q2 + C20Q2 + O((|qII + Q2)5/2)• (6.8)

In a neighborhood of the trajectory of unperturbed periodic motion the quantity Q1 can have any sign and q2 is nonnegative; on the unperturbed motion q1 = q2 = 0. The function (6.8) is analytic in Q1, |q2|1/2 and periodic in

According to the Arnold-Moser theorem, if the quantity

D = C20 A2 — en A + C02 (6.9)

is different from zero, then stability with respect to the variables Q1, q2 takes place (which in the problem of stability of periodic motion means their orbital stability).

Calculations have shown that the following representation holds for the quantity (6.9) corresponding to periodic motions of the kth type:

2(-l)fc4P1 + V2fo(l-fo)P2 ^ -Y2Q1Q2-+ °(£) (k = l,2). (6.10)

Here the following notation is used:

Pi = 1081672704r013 — 8545886208r012 + 29641863168^ — 58187336256^° + + 67406827536r0 — 38350423944r§ — 8059279372r7 + 32358890456r0 —

— 25287068963r0 + 9140543235r0 — 907382250r0 — 418637250r2 + + 137345625r0 — 11390625,

P2 = 3809369088r013 — 34805002752r12 + 139403089152r011 — 324775588608r10 + + 493040181744r9 — 521861998968rg + 405464611788r7 — 234975618790r6 + + 90108662031r0 — 8242961049r0 — 12324007230r0 + 5660803800r2 —

— 305704125r0 — 193640625,

Q1 = (5184r0 — 15552r5 + 15552r4 — 3896r0 — 2079r0 + 630r0 + 225)2,

Q2 = 81r0 — 242r0 + 225.

When k = 1, the coefficient at e2 in the expression (6.10) vanishes only in two cases: when r0 = r0 = 0.557 and when r0 = r0' = 0.877. (The corresponding values are 5 equal to —0.103 and 0.040). Therefore, according to the Arnold-Moser theorem, for values r0 that are not equal to r0 and r0', periodic motions of the first type are orbitally stable if e is sufficiently small.

In the case of periodic motions of the second type (k = 2), the coefficient at e2 in (6.10) is zero at r0 = r* = 0.017 and at r0 = r** = 0.794 (5 = 1.828 and 5 = —0.702). If r0 = r0 and r0 = r**, then orbital stability takes place at sufficiently small e.

The problem of stability of periodic motions for values of r0 equal to r0, r0 (for k = 1) and r*, r** (for k = 2) is left open. Solving this problem requires that the coefficients of terms higher than the fourth degree in |£>1|1/2, o2/2 be calculated in the expansion (6.8).

7. On conditionally periodic motions of a complete system

For values of 5 and h from regions g1 and g2 (Fig. 3), the motions of an approximate system with the Hamiltonian function F1 + F2 are oscillations. We examine these oscillations first in an approximate system and then in a complete system with the Hamiltonian function (3.2). We consider only the case of exact resonance, assuming that the dimensionless inertial parameter a in the Hamiltonian function (1.2) is exactly equal to 2/3 and hence the quantity 5 in the functions (4.2) and (4.4) is equal to zero. The phase portrait of the system with the Hamiltonian function (4.4) is shown in the right panel of Fig. 4.

Admissible values of h are given by the inequality —14.231 < h < 24.107. If h lies in the interval —14.231 < h < 20, then the phase trajectories encircle the equilibrium point of the second type q0 = 0, p0 = —0.716 (%0 = n, r0 = 0.256). If 20 < h < 24.107, then the trajectories encircle the equilibrium point of the first type q0 = 0, p0 = 1.292 (%0 = 0, r0 = 0.835). The value h = 20 corresponds to the separatrix (dashed line in Fig. 4) separating these two sets of closed trajectories.

1. Let d(k denote the frequency of small linear oscillations of the model system in a neighborhood of the equilibrium of the kth type (k = 1,2). Calculations show that d(1) = 20.879, d(2) = 56.738.

(k)

The frequency Q1 of nonlinear oscillations of the model system will be a function of h. To calculate it, we write one of the equations of the system with the Hamiltonian function (4.2)

in the form

d,T

dr

9T

dx

(7.1)

Substituting the expression cos x, obtained from the integral (4.5), in terms of r and h,

\/2(15 - 7r - e) ,----

cos x = —-. e = v 81r2 - 162r + 225 - Ah,

8\/r(l - r)

and taking into account the symmetry of phase trajectories relative to the axis q = 0, we obtain the following expression for period T of nonlinear oscillations:

' max

T = 2V2 J

dr

e-\J40r - 2h + (15 - 7r)e - e2

(7.2)

The quantities rmin and rmax are the minimal and the maximal values of the variable r in the phase trajectory. For a given h they are the real positive roots of the fourth-degree equation:

648r4 - 1944r3 + 8(437 - 4h)r2 - 8(225 + h)r + h2 =0.

(k)

If we take v as an independent variable, then the expression for frequencies can be written as

Qifc) = §f = §d{k)s{k) (fc = 1'2)' (7"3) where s(k) is a function of h. Their graphs are presented in Fig. 7.

-14.231

20 24.107

Fig. 7. Graphs of the functions s(k) (h) (k = 1, 2) from the expressions (7.3) for frequencies of nonlinear oscillations.

2. To investigate nonlinear oscillations in a complete system with the Hamiltonian func-

tion (3.2), it is convenient to use canonically conjugate variables Ij, Wj (j = 1,2) which are

action-action variables in an approximate system with the Hamiltonian function F\ + F2. In this system the coordinate X2 is cyclic, and hence I2 = R2. The quantity Ii is equal to the area,

r

divided by 2n, of a part of the phase plane lying inside a closed phase curve that encloses an equilibrium point of the first or second type:

It can be seen from these equalities that the quantity h is a function of the ratio Ii/I2:

h = *(Ii/I2). (7.4)

It follows from (3.3), (3.4), (4.2)-(4.5) that in the variables Ij, Wj the Hamiltonian function F\ + F2 of the approximate system does not depend on the angle variables Wj and can be written as

H * = Ho(h)+e2 H2(Ii ,I2), (7.5)

where

Hoih) = h, H2{h,I2) = + ± jf Wi/h). (7.6)

The first frequency Q^ of the approximate system (see (7.3)) can also be calculated by the formula

= = (k = 1,2), (7.7)

where ^' is the derivative of the function ^ with respect to I1/I2.

(k)

The second frequency Q2 of oscillations of the approximate system is defined by

^ = f^ = 1 + fi [2h{XV ~ 26) " {k = 2)" (7"8)

(k) (k)

If the initial conditions are such that the ratio Q1 : Q2 is not a rational number, then the motion in the approximate system will be conditionally periodic (of the kth type) with frequencies Q1k) and Q^ (k = 1,2).

3. The Hamiltonian function (3.2) of a complete system in the variables Ij, Wj is given by

H = H0 (I2) + e2H2(h,l2) + £4HA(I1,I2,W1,W2; e). (7.9)

It is analytic in all its variables and 2^-periodic in w1, w2. In the limit as e = 0, the Hamiltonian function (7.9) depends only on the action variable I2. Therefore, in the system realize the case of proper degeneracy [17].

The following inequalities hold for the function (7.9):

H*0- ("0)

The first two inequalities follow immediately from (7.6), (7.7) and (7.3) (see also Fig. 7). Let us check how the third inequality is satisfied.

From (7.7) and (7.3), (7.4) we have the following chain of equations:

d2H2 = = fhj(k)ds^}_ = fhd(k)d^dh = e2Sk)2 jk) ds(fc)

Oil 36 d,h 36 ' d,h 36 ' dh dh 36 dh '

-14.231 0 20 24.107 h

1 -0.02- -1-1- i i i i —1 1

dh i i i i

-0.04-

i dh

-0.06- 0

Fig. 8. Graphs of the derivatives ds(k)/dh (k = 1, 2).

Since the quantities d(k) and s(k) do not vanish, the third inequality of (7.10) is equivalent

to the difference from the zero of the derivatives ds(k)/dh (k = 1,2). These derivatives were

found numerically using the equation s(k) = 2n/(d(k) T) following from (7.3), where T is the

integral (7.2) depending on the parameter h. It turned out that both derivatives are different

from zero (are zero). Their graphs are shown in Fig. 8.

Since all three inequalities (7.10) are satisfied, the motion of the complete system [17, 20]

(k) (k)

with the Hamiltonian function (3.2) is conditionally periodic with frequencies Q1 ' and Q2 (k = 1,2). Only a portion of order exp(—a1e2), where a1 = const > 0, of the phase space is not filled with conditionally periodic trajectories. The quantities Ij (j = 1,2) are close to their initial values for all v:

\Ij(v) — Ij(0)| < a2£2 (a,2 — const).

References

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