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7Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 1, pp. 59-84. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200106
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70E17, 70E40, 70E50, 70H14
On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point
O. V. Kholostova
The motion of a heavy rigid body with a mass geometry corresponding to the Hess case is considered. The suspension point of the body is assumed to perform high-frequency periodic vibrations of small amplitude in the three-dimensional space. It is proved that for any law of vibrations of this type, the approximate autonomous equations of the body motion admit an invariant relation (the first integral at the zero level), which coincides with a similar relation that exists in the Hess case of the motion of a body with a fixed point. In the approximate equations of motion written in Hamiltonian form, the cyclic coordinate is introduced and the corresponding reduction is performed. For the laws of vibration of the suspension point corresponding to the integrable cases (when there is another cyclic coordinate in the system), a detailed study of the model one-degree-of-freedom system is given. For the nonintegrable cases, an analogy with the approximate problem of the motion of a Lagrange top with a vibrating suspension point is drawn, and the results obtained earlier for the top are used. Some properties of the body motion at the nonzero level of the above invariant relation are also discussed.
Keywords: Hess case, high-frequency vibrations, integrable case, reduced system, Lagrange
top
1. Introduction
The existence of a particular integral (an invariant relation at the zero level) of the Euler-Poisson equations of motion of a heavy rigid body around a fixed point was discovered by Hess in 1890 [1] and, somewhat later, by Appelrot [2]. Zhukovskii [3] gave a geometric analysis of motion and proposed a model of a rigid body with Hess mass geometry. An analytical study was carried out in the paper [4], where the system of equations of motion of the body in the Hess case was reduced to (one) Riccati equation; this equation was obtained in a different way in the monograph [5].
Received August 27, 2019 Accepted October 21, 2019
This work was carried out within the framework of the state assignment (project No. 3.3858.2017/4.6).
Olga V. Kholostova kholostova_o@mail.ru
Moscow Aviation Institute (National Research University) Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993 Russia
By now, a number of cases of the existence of the Hess integral or its generalization have been discovered in various problems of solid mechanics and related problems. Particular Hess integrals have been found in the problem of the motion of a gyrostat around a fixed point [7]; the problem of the motion of a heavy rigid body suspended on a weightless rod [8]; for a body moving along a smooth horizontal plane [9]; in the Kirchhoff equations of the motion of a rigid body in an ideal fluid (the Chaplygin case) [10, 11], in the Poincare - Zhukovskii equations of motion of a rigid body with cavities filled with fluid [6]. Two cases of the existence of the Hess integral in the interaction of two force fields are described in [12].
The general conditions for the existence of a Hess type integral in the problem of the motion of a rigid body around a fixed point in a generalized potential field, as well as in the problem of the motion of a rigid body on a smooth plane, are obtained in [13]; here, conditions are given for the existence of a similar integral in the problem of the motion of a gyroscope in gimbals. The paper [14] considers the Suslov problem on the motion of a rigid body around a fixed point in the presence of a nonholonomic constraint; conditions are obtained under which the equations of motion of the body admit the Hess type first integral; in this case, the problem is shown to be isomorphic to the classical Hess case in the Euler-Poisson equations.
The investigation of the Hess case of motion of a heavy rigid body with a fixed point can be reduced to the study of a vector field on a two-dimensional torus. In the series of papers [15-18], some topological properties of the phase flow of the problem are studied in the presence of perturbation (in the class of the Euler-Poisson system). In particular, limit cycles are considered near a critical circle case. The neighborhood of a saddle point in the presence of perturbations splitting the separatrices and chaotic behavior of the system are also discussed.
In [19], it is established that limit cycles on the torus (for the classical Hesss problem) exist only at a zero value of the rotation number. For the generalized Hess case on a pencil of Poisson brackets, the effect of quantization of the rotation number, which can take arbitrary integer values depending on the parameter, is discovered. The presence of a cubic perturbation in a specific Hamiltonian system is shown to lead to a general situation (the Cantor ladder).
The purpose of this paper is to study the dynamics of a rigid body with the Hess mass geometry in a gravity field under the assumption that the suspension point of the body performs high-frequency vibrations of small amplitude in the three-dimensional space. Previously, the approximate autonomous equations of motion of a body with an arbitrary mass geometry in the presence of vibrations of the suspension point of this type were obtained in [20, 21]. These equations have the form of the modified Euler-Poisson equations or Hamiltonian equations, in which the vector of a vibration moment or the corresponding vibration potential is additionally taken into account.
In our paper, we will prove that the vibration moment acting on a body with the Hess mass geometry is always orthogonal to the radius vector of its center of mass relative to the suspension point. From this fact, it is easy to establish that the approximate system of equations of the body motion admits an invariant relation which coincides with the Hess integral for a body with a fixed point. This means that in the approximate problem of the dynamics of the body with a vibrating suspension in the Hess case (as well as for a body with a fixed point) there is symmetry with respect to rotations, and a cyclic coordinate (at the zero level of the integral) can be introduced, which leads to a reduced system with two degrees of freedom.
If there is another cyclic coordinate in this system (the case of vibration symmetry with respect to the vertical), the reduced system can be integrated. In this paper, the corresponding model system with one degree of freedom is investigated in detail. A geometric interpretation of
the motion of the body axis containing its center of mass is given. In the absence of the second cyclic coordinate, the reduced system is apparently not integrable.
Note that the Hamiltonian of the reduced system with two degrees of freedom coincides with the Hamiltonian in the approximate problem of the dynamics of a Lagrange top with a vibrating suspension point, with a zero angular velocity of proper rotation of the top. This analogy is known for bodies in the Lagrange and Hess cases with fixed suspension points. The problem of the motion of the Lagrange top with a vibrating suspension was investigated in various formulations in [22-27]. Taking into account the analogy of two problems, some results obtained earlier in the dynamics of the top are used in the Hess case study.
Some features of the particular motions (stationary rotations around the vertical) of the body in the Hess case in the presence of vibration symmetry relative to the vertical are also discussed under the assumption that the above-mentioned invariant relation does not hold.
2. Problem statement. Equations of motion
Consider the motion of a heavy rigid body, assuming that one of its points O (referred below as the suspension point) performs a specified motion in the three-dimensional space. Let the radius vector O*O have projections u(t),v(t),w(t) in a fixed coordinate system O*XYZ, with the axis O*Z directed vertically upwards. We consider these functions to be periodic (with a frequency Q) and twice continuously differentiable, with zero time average values.
Let us introduce a coordinate system OXYZ obtained from the system O*XYZ by trans-lational motion, and the body-fixed system Oxyz formed by the body's principal inertia axes for the point O. The unit vectors of the axes OX, OY, OZ and Ox, Oy, Oz are denoted, respectively, by a, 3, j and e\, e2, e3. Let m be the body's mass, A, B, C its principal moments of inertia for the point O, with A > B > C. The coordinates of the center of mass G of the body in the body-fixed system are denoted by xG, yG, zG.
Let the maximum length a of the radius vector O* O be small compared with the length £ = OG, and the oscillation frequency Q be large compared with the characteristic frequency -s/gji, where g is the acceleration of gravity. We introduce a small parameter t and the dimensionless frequency according to the formulae
t2 = a/£, g/(m2) = e4w2, w* = s/gl/aQ ~ 1.
In [20, 21], an approximate autonomous system of differential equations describing the body motion relative to the point O was obtained by perturbation theory methods. It represents the modified system of Euler-Poisson equations and has the form
dM/dt = lu + u x Iu = mg + mv, (2.1)
a = a x u, ¡3 = 3 x u, j = j x u. (2.2)
In Eqs. (2.1), (2.2) and further, the symbol d/dt and the dot denote the time derivative in the coordinate systems OXYZ and Oxyz, respectively, M = Iu is the angular momentum of the body, I = diag(A, B,C); u is the angular velocity vector of the body, with the projections p, q,, r onto the body-fixed axes; mg = mg j x OG is the moment of gravity, and mv is the vibration moment defined by the formula [20, 21]
mv = (mVo x (OG x S)). (2.3)
Hereinafter, the angular brackets denote the time averages of the functions contained in them.
The vector Vo in (2.3) is the suspension point velocity. In the coordinate systems O*XYZ and Oxyz its components will be denoted by VX, VY, VZ and vx, vy, vz, respectively, where
VX = du/dt, VY = dv/dt, VZ = dw/dt; vx = Vo • e\, vy = Vo • e2, vz = Vo • e3.
The components of the vector ó in the coordinate system Oxyz are calculated by the formulae
Sx = ~X^ZGVy ~ VgVz^ = ~ zGi'x), Sz = %{vgvx ~ xGvy). (2.4)
The motion of the body relative to the point O can also be described using the approximate autonomous canonical Hamiltonian equations. For this purpose, the vibration potential n^ is added to the potential energy, which is defined by the formula
H V = -2{A52X + B52y + C5l). (2.5)
Further, we will study body motions within the framework of the approximate system written in the form of Eqs. (2.1), (2.2) or in the form of the relevant Hamiltonian equations. An estimate of the accuracy of solutions of these approximate equations (as compared with the solutions of the complete nonautonomous systems) and the corresponding time interval are given in [20, 21].
The subject of this paper is the study (within the framework of the above systems) of the dynamics of the body, the mass geometry of which satisfies the Hess case:
yG = 0, (B - C)Ax% = (A - B)Cz2G. (2.6)
Without loss of generality, we will further assume that xG > 0, zG > 0. This can always be achieved by redirecting the axes of the body-fixed system.
Conditions (2.6) imply that the center of mass of the body lies in the principal inertia plane Oxz, and the radius vector OG of the center of mass relative to the suspension point is perpendicular to one of two circular sections of the gyration ellipsoid
x2 y2 z2 T + ^ + ^ = const,
which contains the axis Oy of the middle moment of inertia [3].
It is well known that in the Hess case the system of equations of motion of a body with a fixed point admits an invariant relation (the first integral at the zero level) of the form
M • OG = 0, (2.7)
which allows one to perform a reduction of the system of equations of motion and to obtain a solution to the reduced system in terms of quadratures.
In this paper, we prove (Section 3) that for arbitrary vibrations of the suspension point (within the framework of the assumptions made), the approximate system of equations of the body motion with the Hess mass geometry also admits an invariant relation of the form (2.7). In Section 4, an integrable case is investigated, the properties of the corresponding model system with one degree of freedom are studied in detail, and a geometric interpretation of the motion of the body axis is given. Section 5 discusses the properties of certain particular body motions (families of equilibria and stationary rotations) in nonintegrable cases.
3. The existence of an invariant relation
3.1. Transformation of the vibration moment and vibration potential
If the body mass geometry satisfies conditions (2.6), then the formulae for the vibration moment and the vibration potential can be transformed to a form more convenient for further study.
Consider first the vibration moment. We expand the double vector product from the formula (2.3)
mv = m(OG(Vo ■ S) - S(Vo ■ OG)) (3.1)
and transform the scalar product VO ■ S taking into account conditions (2.4) and (2.6):
m m , . m
-jZGVxVy + —Vy(xGvz - zGvx) - —c
B-A C-B
mvy
-ZGVx + ,,,, xGvz
AB G x BC
C — B xG , , C — B xG
-BC~-^xG + VzZG) =
After that, relation (3.1) can be rewritten as
mv = m{(V0 • OG) <5x), ö± = rnvy^--^-^OG - Ö. (3.2)
BC zg
Further, note that the components of the vector ¿i with regard to Eqs. (2.4) and (2.6) can be represented in the form
mzGVy m mxGVy
t>lx =--^Ht>ly = - — {XGVZ - ZGVx), diz = —^H-,
and therefore, = (m/B) OG x Vo. Thus, equality (3.2) is reduced to
2
m 2
mv = —((OGV0)OGxV0). (3.3)
B
Hence, it follows that the vector mv of the vibration moment (as well as the vector mg) is orthogonal to the radius vector OG.
We now consider Eq. (2.5) for the vibration potential. Rewrite it in the form
m2 m2 m2 = Tj((ZGVy)'2} + -7^{(XGVZ ~ ZGVX)'2} + -^{(XGVy)'2}
taking into account the formula (2.4) and the condition yG = 0, We exclude the term vy from this expression using the relation v^ + v^ + v2 = Vq (t) where Vq (t) is the square of the suspension point velocity (and (Vj(t)) = const). Discarding the nonessential additive constants, we represent the transformed formula for n as follows:
= - ^A+^c)++ XGVz ~ ZGV't)2)-
We can regroup further summands in angle brackets as
_ m2/(B-A_2 x2G\ 2 fC-B 2 £|\ 2 xGzG 2 \\ AB C J^H BC Xg~ A)lK~~2 B VxVz
The second condition in (2.6) implies that the relations in the two parentheses are equal to x2G/B and z2/B, and the expression in the angle brackets is represented by a complete square. We get in the end
2 2
n. = -^{(w* + ^)2) = ~((OG • Vof). (3.4)
3.2. Invariant relation for Eq. (2.1)
We now prove that equality (2.7) is an invariant relation for Eq. (2.1). We differentiate the expression M ■ OG with respect to time, taking into account Eq. (2.1) and the orthogonality of the vectors mg and mv to the radius vector OG:
dV ' dt dt
= (mg + mv) ■ OG + M ■ (w x OG) = M ■ (w x OG).
The product M ■ (w x OG) can be written in terms of the components of the multiplied vectors, with regard to conditions (2.6):
M ■ (w x OG) = [(A - B)zgv + (B - C)xGV\q =
(B - C)Xg~,,~ , (B - C)xg ~r n /r ^^
v ' -q(ApxG + CrzG) = -— ' q (M ■ OG).
Czg Czg
The result is
jf(M ■ OG) = {B c^XGq(M ■ OG). (3.5)
It follows from this that relation (2.7) is an invariant of Eq. (2.1). Thus, we have proved the following theorem:
Theorem. If the body mass geometry corresponds to the Hess case, then Eq. (2.1) admits the invariant relation (2.7) for any vibration features of the body suspension point determined by the vibration moment (3.3).
3.3. Reducing the order of the system of equations
The existence of an invariant relation in the problem under study (as in the case of a body with a fixed point) is due to symmetry with respect to the rotation of the body around the axis OG, which is perpendicular to the aforementioned circular section of the gyration ellipsoid. This circumstance allows one to introduce a cyclic coordinate (at the zero level of the impulse) and to reduce the order of the system. For a body with a fixed point in the Hess case, this procedure is described, for example, in [6].
Let us turn to the system Ox'y'z' of nonprincipal inertia axes, with unit vectors ei, e'2, e's, by rotating the principal inertia plane Oxz around the Oy axis at an angle
a = arctan — = arctan J
zG V (B - C)A
As a result, the Oz' axis passes through the center of mass G, and x'G = y'G = 0, z'G = OG = i.
We will define the orientation of the body-fixed system Ox'y'z' in the OXYZ-system by the Euler angles d, ^ introduced in the usual way.
3.3.1. Transformed Euler Equations
Let us project Eq. (2.1) onto the axis of the coordinate system Ox'y'z': hp + hsf + (Is - h)qr + Iispq = mgi^2 + mvx>,
hq + (Ii - Is)rp - hs(p2 - r2) = -mgi^i + mVy>, (3.6)
Isr + IisP + (I2 - Ii)pq - Iisrq = mvz<.
In Eqs. (3.6), the projections Yi = sin d sin y2 = sin dcos ys = cos d of the vectors y as well as the projections mvx/ ,mvy/ ,mvz/ of the vibration moment vector onto the axis Ox', Oy', Oz' are used. The introduced moments of inertia are calculated using the formulae I2 = B and
Ii = A cos2 a + C sin2 a, Is = A sin2 a + C cos2 a, Iis = (A - C)sina cos a. (3.7)
Note that Iis > 0, due to the assumptions made.
Given the second condition in (2.6), we find that
A = IiIs - Ifs = I2ls. (3.8)
Then, taking into account the triangle inequality for axial moments of inertia, we have
I2s = Is(li - I2) <Is2. (3.9)
Further, we note that the expression (3.3) for the vibration moment is written in vector form and, for a fixed axis Oy = Oy', does not depend on the choice of the two other axes of the body-fixed system. Setting the components of the multiplied vectors in the coordinate system Ox'y'z', we obtain
m2i2
mv = ——{-(vy>vz>), {vx/vz,),0), (3.10)
where
Vx' = Vo ■ e'i, vy = Vo ■ e'2, Vz' = Vo ■ e's. The third equation of the system (3.6) can be rewritten with regard to equality (3.8) in the
form
I2
hs'P + hf = q(hsP + hr), (3.11)
Is
whence we obtain the invariant relation (2.7), which reduces to the condition Mz' = 0 and has the form
Iis p + Is r = 0. (3.12)
Using this equality, we exclude the variable r from the first two equations of the system (3.6). Then, taking into account relation (3.9), we rewrite these equations in the form
P + ~rPQ = -r(mg£j2 + «w), q ~ ~rP = ~r{-mghi + nivy'), (3.13)
where the quantities mvx/ and mvy/ are defined in the formula (3.10).
3.3.2. Hamiltonian of the system
We now obtain the Hamiltonian of the system. The expression (2.5) for the vibration potential can be rewritten with regard to the condition OG = £e'3 in the form
m2f2 m2C2
it = -~nr((Vo ■ 4)2) = --77t Sin 4> sin 9 - Vy cos 0 sin 9 + cos 9)2). (3.14) 2B 2B
From the expression (3.14) it follows that the vibration potential depends on six quantities that determine the type and properties of vibration, the average values of the squares of the components (in the coordinate system OXYZ) of the velocity vector Vo and their combined products:
ax = (VX), aY = (V2), az = (V2),
(3.15)
axY = (Vx Vy ), axz = (Vx Vz), aYz = (Vy vz ).
These constant quantities are parameters of the problem. By turning the axes O*X and O*Y one can always achieve the condition axY = 0.
With these notations, the expression (3.14) can be rewritten as
m2f>2
= ———— [(a-i sin2 ip + a-2 cos2 ip) sin2 9 + {axz sin 0 — ayz cos ip) sin 29]. (3.16)
2B
Additive constants are omitted here and two more notations are introduced as follows:
ai = ax — az, a2 = aY — az. (3.17)
Using the nonprincipal inertia axes, the kinetic energy of the body in the coordinate system OXYZ can be written in the form
T = hp2 + hq2 + hr2) + hipr. (3.18)
The quantities p, q, r satisfy the kinematic relations
p = ip sin 9 sin p + 9 cos y, q = ip sin 9 cos p — i^sin p, r = ip cos 9 + Lp. (3.19)
We introduce the generalized momenta related to the angles p, 9, p by
dT
Pip = -r = L1 sin 9 sin Lp + L'2 sin 9 cos Lp + Ls cos 9,
dip
dT dT
Pe =—r = Li cos ip - L2 simp, p = — = L3. (3.20)
d9 dp
Here
dT dT dT
Li = — = hp + hsr, L2 = — = hq, L3 = — = I3r + hsP- (3.21)
dp dq dr
Solve Eqs. (3.21) for p, q, r and rewrite the expression (3.18) in the form
T = £rL* + + - ^Ls- (3.22)
According to relation (3.8), the coefficients in the terms with Lf and Lf are the same.
We then find the quantities Lk (k = 1,2,3) using the formulae (3.20). After some transformations, we have
f = J-
2/2
9 , iPi>~Pv cos i
Pe + "
sin2 0
Ii 2 Iis (p% - pv cos e)siny - pe sin e cos y + A" ¡tafl ^ (3-23)
The Hamiltonian of the system is calculated by the formula
H = T + ng +nv, (3.24)
where ng = mgicos e, and the expression for nv is given in Eq. (3.16).
The invariant relation corresponding to the system with the Hamiltonian (3.24) has the form pv = 0. Given the third equation in (3.21) and equalities (3.19), we rewrite this relation in the form
(Is cos e + Iis sin e sin y)ip + Iis e cos y + Isy = 0. (3.25)
At the zero level of the impulse pv, the y coordinate is cyclic, and the Hamiltonian reduced at this level is determined by the expression
1 / 2 , p%
H = —lPg + -—- + mgi cos 9 -
2h\ sin (3.26)
m2t2
[(a,i sin2 ip + a-2 cos2 ip) sin2 9 + (axz sin ip — ayz cos ip) sin 29].
2I2
We rewrite the Hamiltonian (3.26) by introducing the dimensionless momenta pi and p2, and the dimensionless time t according to the formulae
p% = hupi, pe = h^p2, t = wot, wo = (mgi/I2 )i/2. As a result, we obtain
1 / „2
r - ' ' "2
(p22 + -Eh) +COS0-V sin2 0 J
n sin e, (3.27)
mi
-[(a,i sin2 ip + a-2 cos2 ip) sin2 9 + (axz sin ip — ayz cos ip) sin 29].
2I2g
The reduced Hamiltonian (3.27) (or (3.26)) describes the motion of a two-degree-of-freedom system, which will also be called reduced. To the equations of motion of this system we add the (dimensionless) equation for the angle y, which follows from relation (3.25):
y' = -(asin e sin y + cos - a&cos y. (3.28)
Here we have introduced the notation a = Iis/Is, with 0 < a < 1 due to inequality (3.9); the prime means differentiation with respect to t.
Note that Hamiltonian r coincides with a similar Hamiltonian describing (in an approximate autonomous problem) the motion of a Lagrange top with a vibrating suspension point [26], at zero constant of the cyclic integral corresponding to the proper rotation angle. For bodies with the Lagrange and Hess mass geometry, when they move around their fixed points in the gravity field, a similar analogy is well known (see, for example, [6]), and the relevant reduced system can be integrated by quadratures.
For arbitrary values of the vibration parameters, the reduced system with Hamiltonian (3.27) is apparently not integrable. Integration of the system can, however, be carried out if there is a symmetry of the vibrational field with respect to the vertical. This case will be discussed in the next section.
4. Integrable case
Let the vibration parameters of the problem at hand satisfy the relations as follows:
ai = a2 (ax = aY), axY = axz = aYz = 0. (4.1)
This symmetric case is realized, for example, if the suspension point of the body moves according to the law (l1 and l2 are constants)
u(t) = el1 cos Qt, v(t) = el1 cos 2Qt, w(t) = el2 sin Qt.
If conditions (4.1) hold, then in the system with Hamiltonian (3.27) there is another cyclic coordinate, p (as in the case of a body with a fixed point), and p1 = c = const. The Hamiltonian of a corresponding one-degree-of-freedom system (hereinafter referred to as the model Hamiltonian) has the form
1 2 c2 1 2 ml
ri = o \P2 + —T7 +cosfc> - -o-cos 9, a = —{az-ax)- (4.2)
2 \ sin2 9 J 2 +2g
The dimensionless parameter a characterizes the ratio between the intensities of the vertical and horizontal components of vibration, and can take values of any sign. Further, we assume that a = 0.
Consider the energy integral r1 = h = const of the model system. We write it, given that p2 = 9', in the form
c2
9'2 =2h-2cos9 + acos29--(4.3)
sin2 9
Equation (4.3) can also be represented in algebraic form by introducing the variable u = cos 9 (|u| ^ 1):
u'2 = f (u), f (u) = (1 — u2)(2h — 2u + au2) — c2. (4.4)
The regions of possible motions of the system are determined by the condition f (u) ^ 0. In these regions we obtain the quadrature
u uo
For a = 0, the function f (u) is a fourth-degree polynomial in u, so solutions of Eq. (4.5) are written in terms of elliptic functions. If u = u(t) is a solution of Eq. (4.5) and 9 = 9(t) is
the corresponding change in the nutation angle, then the change in the angles of precession is determined by quadrature from the equation
C
$ = —V (4-6)
sin2 6
which completes the integration of the reduced system with the Hamiltonian (3.27) under conditions (4.1). After that, it remains to integrate the nonautonomous first-order differential equation (3.28).
Further, we also need Eqs. (3.13), rewritten with regard to conditions (4.1). We calculate the components of the vibration moment vector (3.10) under these conditions. Then we move to dimensionless time t, make dimensionless the projections p and q of the angular velocity with the help of the factor w0 and, retaining the same notation for these quantities, we represent Eqs. (3.13) in the form
p' + apq = (1 - CT73)Y2, q' - ap2 = -(1 - )Yi. (4.7)
To Eqs. (4.7) we add the Poisson equations for the values y1 , y2, Y3. After excluding the variable r using Eq. (3.12), the Poisson equations take the following dimensionless form:
Yl = -aY2P - Y3q, y2 = (aYi + Y3)p, y3 = Yiq - Y2'p. (4.8)
The system of equations (4.7) and (4.8) has three first integrals, namely, energy and area integrals, and a geometric integral, of the form
\(P2 + q2) + 73 - |y32 = const, (4.9)
PY1 + qY2 = const, (4.10)
Yl2 + Y22 + y2 = 1. (4.11)
We now consider in more detail the body motion in the integrable case under study.
4.1. The case c =0
First let c = 0. Relation (4.6) yields that ^ = const in this case, and the trajectory of the apex (the end of the vector e'3) on the Poisson sphere lies on one of the circles passing through the upper and lower poles of the sphere.
The axis Oz' of the body performs pendulum-type motions described by the canonical equations with a model Hamiltonian
rio = ^P22 + cos Q-X-o cos2 Q. (4.12)
When 6 is replaced by n - 6, this Hamiltonian coincides with the Hamiltonian function describing the pendulum-type motions of the Lagrange top with a vibrating suspension point, considered in [27]. We briefly describe the motion of this model system.
4.1.1. Motion of the model system
The particular solutions p2 = 0 and 6 = 0 or 6 = n of the system with the Hamiltonian (4.12) correspond to the relative body equilibria for which the radius vector OG is in a vertical position (upper or lower poles of the Poisson sphere). The upper equilibrium is stable
when the condition a > 1 is fulfilled (or, in the original notation, when az — Bg/(m£) > ax) and is unstable otherwise. The lower equilibrium is stable for a ^ —1 (aZ + Bg/(m£) > ax), and unstable otherwise.
In addition, for |a| ^ 1, the system with the Hamiltonian (4.12) has also two symmetric equilibria p2 = 0, 9 = 9* = ± arccos (1/a) corresponding to the positions of the radius vector OG that are inclined to the vertical; they are stable if a ^ —1 (aX ^ aZ + Bg/(m£)), and unstable if a ^ 1 (ax < az — Bg/(m£)).
The equilibria considered correspond to the energy levels h0 = 1 — a/2, hn = —1 — a/2 and h* = 1/(2a) of the system.
The regions of other possible motions of the system are given by the square inequality 2h — 2u + au2 > 0, which follows from the condition of positivity of the function f (u) from Eq. (4.4) at c = 0. We introduce the notation for the roots of the above square trinomial as
U\
1 - VI - 2ha
a
U2
1 + VI - 2ha
a
The phase portraits of the system with Hamiltonian (4.12) are shown in Fig. 1a, 1b, and 1c for the cases a < —1, —1 < a < 1, and a > 1, respectively. The corresponding regions with different types of the system motion are shown in Fig. 2 in the parameter plane a, h.
In region 0 in Fig. 2 motions of the model system are not possible. In region 1, given by the conditions a < —1 and h* < h < hn, oscillations in the vicinity of the lateral equilibrium 9 = 9* (Fig. 1a) occur in the system, with the frequency wi written below.
(c)
Fig. 1. Phase portraits of the model system at c = 0.
h '
4 2 -1 \ 1 5 \ \ 3
-/— _ 1 n 1 o
1 - 0 U -1 -2" 12 2
Fig. 2. Regions with different types of the motions of the model system at c = 0.
When hn < h < h0 (region 2), we have oscillations, with the frequency w2, in the vicinity of the lower equilibrium d = n. For 0 < |a| < 1 they are all oscillations of this type (Fig. 1b), for a > 1 they are part of such oscillations (with energy level h <h0, see Fig. 1c), and for a < —1 they are oscillations corresponding to the closed curves covering three singular points of the system in Fig. 1a.
For the values a > 1 and h0 < h < h* (region 3), two types of motions occur in the system, with the same frequency they are one of the oscillations in the vicinity of the lower position d = n (with energy level h > h0) and one of the oscillations in the vicinity of the upper position d = 0 (Fig. 1c).
Finally, when a < 1, h > h0 or a > 1, h > h*, rotations occur in the system (see Figs. 1a-1c); the rotation frequencies are equal to w4 and in regions 4 and 5 of the part of the parameter plane at hand, lying, respectively, to the left (with a = 0) and to the right of the dotted curve.
The frequencies introduced are calculated by the formulae as follows [28, 30]:
<* = M5 {i = 1.....4)' = (413)
where K(ki) is the complete elliptic integral of the first kind, and
= -\/M(l-ui)(l+u2), 6 = y/\a{u2 ~ui)\, 6 = 6, 6 = VaMN,
/ 2= 2(m2-m) 2= (l-u2)(l + Ul) 2 = 4 -(M-AQ2
1 (1-Ul)(l+U2y 3 (l-u1)(l + U2), 5 4M7V '
M2 = (1 — m)2 + n2, N2 = (1 + m)2 + n2, m = 1/a, n2 = (2ha — 1)/£2.
The quantities 6 and k4 are obtained from 6 and ki by replacing m o u2, and k2 = k-1.
The oscillations and rotations considered are expressed in terms of elliptic functions. Exceptions are the cases h = h0, h = hn and h = h*, corresponding to the equilibrium points and one of the oscillations or asymptotic motions; because of the presence of a multiple root of the function f (u)), these motions are defined via elementary functions. We do not write the explicit relations d = d(r) on the trajectories of the model system; they can be obtained using the formulae from [28, 30].
4.1.2. On body motions at c = 0
The motion of the body relative to the axis Oz' with ^ = const is described, according to Eq. (3.28), as follows:
p = —a9' cos p. (4.14)
In particular, for the equilibrium values 9 = 9o = const of the model system with the Hamiltonian (4.12), Eq. (4.14) yields p' = 0. We obtain the families of equilibria of the body in the coordinate system OXYZ, when angle p takes arbitrary values. Note that, in view of the analogy mentioned above, these families of equilibria correspond to the stationary rotations of the Lagrange top around the vertical (inverted and hanging top) and inclined axes studied in [25]. Let's solve the stability problem of the found body equilibria. Put in the system (4.7), (4.8)
p = xi, q = X2, Yi = Yio + X3, 72 = 720 + £4, 73 = cos 9o + x5 (4.15)
under the condition 720 + 7^, + 7^, = 1.
We calculate the characteristic equations of the linearized equations of perturbed motion. For the cases 730 = ±1, 710 = 720 = 0 they have the form
A(A2 + a ^ 1)2 = 0,
where the upper and lower signs correspond to the cases 730 = 1 and 730 = —1.
The necessary stability condition of these equilibria (the condition of the absence of roots with positive real parts) is determined by the inequality a ^ 1 ^ 0. The characteristic equation for the case 730 = 1/a is
A3 ( A2 + ~~~~ ) =0.
Taking into account the existence region, we find the necessary stability condition as a ^ —1.
To obtain the sufficient stability conditions, we take the first integrals of the system (4.7), (4.8) and use the Lyapunov stability theorem [29]. For body equilibria corresponding to the values 90 = 0 and 90 = n, we introduce perturbations in relations (4.9) and (4.11) using formulae (4.15), and rewrite these relations in the form
= ^ {x\ + xl) + (1 =f <t)x5 - | xt (4.16)
V2 = x2 + x2 ± 2x5 + X2, (4.17)
where, as above, the upper and lower signs correspond to 90 = 0 and 90 = n. The Lyapunov function is chosen as the sum of squares of the first integrals
V = V2 + V22. (4.18)
It is sign-definite with respect to the variables x1,...,x5 if the function V1 considered on the set of values of xj that make the function V2 vanish is sign-definite. Solve the equation V2 = 0 for x5 as
22 x3 + x4 i
£5 = T 2 + • • •
and substitute this expression in relation (4.16):
Vi|v2=o = ^ (xj + x\) + ^y^ (xj +x24) + ....
Ellipses in the last two formulae mean sets of terms higher than the second degree in x3 and x4.
If the condition a ^ 1 > 0 is fulfilled, then the function V1 obtained in such a way is positive-definite with respect to x1, x2, x3, x4, and the function V is positive-definite in all five variables. Therefore, this inequality is a sufficient condition for the stability of the considered equilibrium positions of the body with respect to perturbations of the quantities p, q, 71, 72, 73.
To study the stability of the body equilibria corresponding to the value 9 = 9*, let us represent the integrals (4.9) and (4.10), with regard to relations (3.19), in the form
i (0/2 sin2 e + er2) + cos e -1 cos2 e = const, (4.19)
$' sin2 9 = const. (4.20)
We introduce the perturbations by the formulae 9 = 9* + x1, 9' = x2, = x3 and rewrite these integrals as
Vi = a2a21 (~axi + x1) + \ x2-> = ®3(sin2 9* + sin 20* Xi + .. .)•
Here, the ellipsis means the set of the terms of the second and higher degrees in x1.
The Lyapunov function is again chosen in the form (4.18). Considering the relation V2 = 0, we find that x3 = 0. Taking into account this condition as well as the existence region of the equilibrium, we conclude that the function V1 is positively defined with respect to the variables x1, x2 if the inequality a < —1 holds. This condition ensures the sign-definiteness of the function V with respect to x1, x2, x3 and represents the sufficient stability condition in relation to these perturbations for the body equilibrium under study.
Thus, the necessary and sufficient stability conditions (with respect to the corresponding sets of variables) of the body equilibria at hand, up to an equal sign, coincide with the stability conditions for the corresponding equilibria of the model system.
Consider now the body motions different from the equilibrium positions. The body axis Oz' performs the pendulum-type motions (oscillations or rotations) described above. From relation (4.14) it follows that under the condition cos p = 0 (p = ±n/2, p = ±3n/2, etc.) there are no body rotations around the axis Oz', i.e., the body moves as a physical pendulum.
If cos p = 0, then Eq. (4.14) can be integrated, and the result is represented as
tan + = Ce~a0 (C = const).
When the solution 9 = 9(t) of the model system is substituted into this equation, we obtain the relation p(t).
If the solution 9 = 9(t) describes oscillations of the body axis, then the angle p (along with 9) varies in a limited range periodically, and its period is equal to the period of oscillation of the body axis.
If the solution 9 = 9(t) corresponds to rotations of the body axis, then, depending on the initial conditions, the angle p tends, as t ^^ and t ^ —<x, to one of the equilibrium values satisfying the condition cos p = 0. For example, for the rotation from the upper half-plane of the phase plane of the model system (9' > 0, see Fig. 1) and the initial value p = p0 from the interval —n/2 < p0 < n/2, we find that p ^n/2 as t ±<.
4.2. The case c = 0
We now consider the general case c = 0.
4.2.1. Motions of the model system
The corresponding model system is described by the canonical equations with the Hamil-tonian (4.2). In contrast to the case c = 0, the body axis Oz' cannot pass through the vertical, the angle 9 varies in the range (0,n).
Equilibrium positions. The equilibrium positions of the system are stationary points of the reduced potential energy
IIi = cos 9 — — cos2 9 +
c2
2 2 sin2 9
satisfying the equation
sin4 9(a cos 9 — 1) — c2 cos 9 = 0.
Let us rewrite this equation in algebraic form with respect to the value u = cos 9 (|u| < 1), presenting it as
c2u + (1 — u2)2
a = G(u), G(u) = (1_\;2)2t;; • (4-21)
The equilibrium values of the model system are the abscissas of the intersection points of the graph of the function y = G(u) and the straight line y = a. Considering the parameter c fixed, we find the number of such points depending on the parameter a. Consider the properties of the function G(u). Its derivative is
_ 4c2-o3 — (1 —u2)3 { ' ~ (1 -1,2)3«2 •
Equating this derivative to zero, we find the equation for the extremum points of the function G(u)
u2 + ciu — 1 = 0, ci = (2|c|)2/3 (ci > 0), (4.22)
which has a unique solution
u = u* = (0 < t/,* < 1)
on the interval (—1,1). This solution corresponds to the minimum point of the function G(u). At this point we have
G(u*) = i[ci(c? + 6) + (c2 + 4)3/2] > 1. (4.23)
8
On the interval —1 < u < 0, the derivative G'(u) retains a constant sign (negative), and the function G(u) monotonically decreases, vanishing at the point u = u** determined by the equation c2u = —(1 — u2)2.
For a fixed value of c, the graph of the function y = G(u) on the intervals (—1,0) and (0.1) is shown in Fig. 3 by the solid lines.
For a < G(u*), the straight line y = a intersects this graph at one point, and for a > G(u*), at three points. In these cases, the model system with the Hamiltonian (4.2) has one and three equilibrium positions, respectively.
To study the stability of the equilibrium positions found, we calculate the second derivative of the function ni (0); subject to condition (4.21), we obtain
n'lgg = u(1 - u2)C'u(u).
For the equilibrium points located on the left (marked with g1) branch of the graph of the function y = G(u), as well as on the increasing part (g3) of the right branch of the graph of the function (see Fig. 3), the inequality n"dd > 0 is fulfilled, and these points correspond to the points of minimum of the potential energy and to the stable equilibria of the model system. The equilibrium points located on the descending part (g2) of the right branch of the graph of the function y = G(u) correspond to the points of maximum of the potential energy (for them n"dd < 0) and to the unstable equilibrium positions of the model system.
The dotted line (marked with g4) in Fig. 3 is the branch of the function y = G(u) on the interval u > 1. It will be needed to determine the total number of roots of the polynomial f (u) from relation (4.4) for integrating Eq. (4.5).
Oscillations and asymptotic motions. Let us describe the motions of the model system different from the equilibrium positions. Consider the inequality f (u) ^ 0 defining admissible regions of motion. Since f (±1) = -c2 < 0 and f (±rc>) ~ -au4, the system motions exist if the graph of the function y = f (u) has one of the types shown in Figs. 4a-4c and 4d for the cases a > 0 and a < 0, respectively. Note that the case similar to that shown in Fig. 4b, when one of the two positive intervals of the function f (u) is located to the left of the interval (-1,1), is not implemented in this problem.
Figure 4 does not show the cases of double roots of the function f (u), which correspond to the equilibrium positions of the model system or extraneous "equilibria" lying outside the interval (-1,1). These cases are boundary cases; when passing through them, the number of zeros of the function f (u) changes, which leads to a qualitative change in the behavior of the model system and/or (in the case of extraneous "equilibria") to a change in the integration technique.
u
Fig. 4. Different forms for the function f (u).
The boundary cases are given by the conditions f (u) = 0, f'u(u) = 0. Two polynomials f (u) and fU(u) have common roots if their resultant in u equal to
Rs(c2) = -16a{16a3c6 - [8a4 + 64a3h + (32h2 - 12)a2 - 72ah + 27]c4 +
Consider the equation Rs(c2) = 0 (with a = 0) as cubic with respect to c2. It can have one or three roots, depending on the sign of the discriminant of the equation equal to
The expression D1, in turn, is a cubic polynomial with respect to h, with a negative discriminant equal to —20155392(4a2 + 1)3. Therefore, for all values of a, the polynomial Di has a unique real root h = h*(a).
If h > h* (a), then D1 > 0, and D < 0; in this case, the polynomial Rs(c2) has a unique real root. If h < h*(a), then D > 0, and this polynomial has three real roots. Only positive roots should be taken into account; they will correspond to real or extraneous (in the above sense) equilibria of the model system.
The cases a > 1,0 < a ^ 1, —1 ^ a < 0, and a < —1 are qualitatively different. The corresponding discriminant curves (at a fixed value of a) are shown in Figs. 5a-5d in the parameter plane c2, h. The points on the h-axis marked in this figures correspond to the energy levels of the equilibrium positions of the model system at c = 0. Some of them are generators for equilibrium curves for c2 > 0; of the rest, the curves going to the region c2 < 0 are initiated.
The symbols g that mark the various equilibrium curves in Fig. 5 correspond to the same symbols on different branches of the graph of the function y = G(u) in Fig. 3. Comparing Figs. 3 and 5 for fixed values of c2 and a, it is possible to determine the equilibrium values of u = u^) (we will mark them with the same index) and the corresponding energy levels h = h^).
+ [a5 + 24a4h + (88h2 - 15)a3 + 2h(48h2 - 31)a2 + + (16h4 - 196h2 + 12)a - 8h(h2 - 9)]c2 + (1 - 2ah)[(2h + a)2 - 4]2}
(4.24)
vanishes.
D = -(a2 + 2ah - 1)2 Di,
Di = 64ah3 - (96a2 + 36)h2 + 12a(21 + 4a2)h - 8a4 - 9a2 - 108.
(4.25)
The point A* in Fig. 5a corresponds to the minimum point in Fig. 3. Its abscissa c2 = c\ is the result of reversing the relation a = G(u*) (the quantity G(u*) is defined in Eq. (4.23)) with respect to c2, taking into account the relation between the quantities c and ci from Eq. (4.22), and the ordinate Ha = h*(a). The point B* corresponds to the case of a double root h = hB = = (1 — a2)/(2a) of the discriminant D from Eq. (4.25) and a double root c2 = c2B = 1/a of the polynomial Rs(c2).
In regions 0 in Fig. 5 motions of the model system are not possible.
For points (c2,h) located in Fig. 5a to the left of the line c2 = cA and above region 0, the phase trajectories of the model system are depicted in Fig. 6a in the plane of the variables 0, p2 (for 0 G (0,^)). The energy level h = h(1) corresponds to the stable equilibrium 0 = 0^) (hereinafter 0^) = arccos u^), i = 1,2,3). For h(1) < h < h(2) (region 1 and part of region 2 below the curve g2), we have oscillations in the vicinity of this equilibrium. The energy level h = h(2) corresponds to one of these oscillations and the stable equilibrium state 0 = 0(2). For the values in the interval h(2) < h < h(3) (region 3), two oscillations occur in the system in the vicinity of two stable equilibrium positions. For h = h(3), we have the unstable equilibrium 0 = 0(3) and asymptotic motion (separatrix in Fig. 6a). For values h > h(3) (part of region 2 above
(a) (b)
Fig. 6. Phase portraits of the model system at c = 0.
the curve g3) oscillations occur in the system, which in Fig. 6a correspond to the closed curves covering all three singular points (equilibrium positions) of the model system.
For points (c2 ,h) located in Fig. 5a to the right of the line c2 = c}A, as well as for points in Figs. 5b-5d (above regions 0), for h = h^) we have the stable equilibrium d = , and for h > h(!) (regions 1, 2 and 4 in these figures) oscillations in its vicinity. The corresponding phase portrait is shown in Fig. 6b.
For the values of the parameters lying inside regions 1, 2, 3, and 4 in Fig. 5, the graphs of the function y = f (u) have the form shown, respectively, in Figs. 4a, 4b, 4c, 4d.
Remark. In the case a = 0 when vibrations of the suspension point are absent, we obtain the following equation of the discriminant curve from the formula (4.24):
27c4 + 8h(h2 - 9)c2 - 16(h2 - 1)2 = 0. (4.26)
The corresponding graphical representation of this case is Fig. 5c, with h0 = —hn = 1. Previously, an equation of a similar curve (written in parametric form and using parameters different from our parameters by constant factors) and a diagram similar to Fig. 5c were given in [19].
Using the notation given in Fig. 4 for the roots of the polynomial f (u), we find [28, 30] that the frequency w1 of oscillations of the model system in region 1 and of both types of oscillations in region 3 are defined using the first formula in (4.13), where
^^^VH^-usKus-ux), =
and K(k1), as before, is the complete elliptic integral of the first kind. The frequency w2 of oscillations in region 4 is calculated by the same formula in which we assume
2 b 2 V (U4-U2)(U3-Ui)'
For points of region 2, the function f (u) has a pair of complex conjugate roots u3,4 = m±i n. Let be M2 = (u2 — m)2 + n2, N2 = (u1 — m)2 + n2, then the frequency w3 of oscillations in this region is determined by the second formula in (4.13) (with replacement of the index 5 by 3), where
On the boundary curves h = h^) (i = 1,2,3), Eq. (4.5) is integrated in elementary functions.
4.2.2. On body motions at c = 0
We now consider body motions corresponding to the solution 9 = 9(t) of the model system. If this solution describes one of the oscillations of the system, then, according to Eq. (4.6), the angular velocity of precession of the body axis changes periodically, with a period equal to the period of this oscillation, and retains a constant sign. This means that the angle ^ increases or decreases monotonically. The trajectory of the apex on the Poisson sphere is a curve enclosed between two parallels corresponding to the limits in variation of the angle 9 for the oscillation under study (Fig. 7a).
Fig. 7. Trajectories of the apex on the Poisson sphere.
For the equilibrium solutions 9 = 90 = const of the model system, the angular velocity of precession of the body axis is constant, according to Eq. (4.6). This motion corresponds to the stationary rotation of the system with the Hamiltonian (3.27) (under conditions (4.1)). The apex trajectory is one of the parallels on the Poisson sphere (Fig. 7b). If the generating solution 9 = 9o of the model system is unstable, then the stationary rotation is also unstable (with respect to perturbations of the quantities 9 and p2), since the characteristic equation of linearized equations of perturbed motion has a positive real root. If the generating solution 9 = 90 is stable, then, based on the Routh theorem on stability of stationary rotations of a holonomic conservative system with cyclic coordinates [31, 32], this stationary rotation is stable with respect to perturbations 9 and p2. And on the basis of Lyapunov's addition to the Routh theorem (see ibid.), it is also stable with respect to small perturbation of the quantity p\.
The body motions is precession in this case, for which the angular velocity of its proper rotation is generally variable, according to the equation
, c(a sin 9o sin y + cos 9o) sin 9o
Under the condition | cot 90| < a, Eq. (4.27) has two constant solutions y = y0 = const (equilibrium points) defined by the equality sin y0 = — cot 90/a, one of them is asymptotically stable and the other is unstable. If cot 90 = ±a, then there is one equilibrium point y0 = ^n/2, and it is unstable.
The solutions 9 = 90, y = y0, = const correspond to stationary rotations of the body around the vertical. Necessary and sufficient conditions for their stability are the simultaneous fulfillment of the conditions of stability and asymptotic stability of the corresponding equilibrium points of the model system and Eq. (4.27). To verify the sufficiency, we choose the Lyapunov function in the form V = V2 + V22 + £2/2, where Vi and V2 are the energy and area integrals on the disturbed motion, and £ = y — y0. By virtue of the equations of motion, the derivative V' = d£2 + O(£3) (d < 0) of the function V is a negative sign-constant function for sufficiently small values of £. Hence, on the basis of the Lyapunov stability theorem, the stability of this motion follows. The necessity of the above stability conditions is examined when considering the characteristic equation of the linearized equations of perturbed motion.
Nonconstant solutions of Eq. (4.27) are determined from the relation
dy
, 3 • ^TAT-to), (4.28)
J a + f sin y sin2 9 v ' v 7
where the notations a = cos 90, f = a sin 90 are introduced, and t0 is an arbitrary constant.
If a2 > f2, then the derivative y' retains a constant sign, and the angle y monotonously increases or decreases. Integrating (4.28), we obtain the equation
cos2[w(t — t0)] — sin2 [w(t — T0) + ¿0]
cos y =__^___^__,
cos2[w(t — t0)] + sin2 [w(t — t0) + ¿0] '
c\/a2 - ^ x + 13 w =-—, do = arctan ■
2 sin2 6q ' ' ' sja2 - (32'
The rotation of the body around the Oz' axis is periodic, with a period equal to n/w. During this period, the body makes a turn of 2n, returning to its starting position. In the case of a2 < f2, the integration of Eq. (4.28) leads to the relation
y _ - t,an(y0/2) + cot, (y0/2) e-Wl(r-ro) t an — —---^-,
2 e-wi(r-T0) — 1
C- Vf32 - a2 + <fo 13- VP2 - «2
w\ =-^-, tan — =-,
sin2 90 2 a
from which it follows that, as t ^ x> and t ^ —x>, the angle y tends to the asymptotically stable and unstable equilibrium points of Eq. (4.27), respectively. If a2 = f2, then we obtain
y 1 ac tan — = =f —-r + 1 , W2 =
2 \W2(t — T0) /' 2sin2 90'
where the upper and lower signs correspond to the cases a = f and a = —f. As t ^ ±cx>, the angle y tends to the equilibrium value y0 = ^n/2.
5. On the dynamics of a body with Hess mass geometry in nonintegrable cases
We make some remarks on nonintegrable cases in the studied approximate problem of dynamics of a body with the Hess mass geometry in the presence of the suspension point vibrations. The reduced system with two degrees of freedom with the Hamiltonian (3.27) is nonintegrable if the vibration parameters do not satisfy conditions (4.1). If these conditions are satisfied, but relation (3.28) is violated, then the problem is also nonintegrable. However, due to the existing symmetry, which is characteristic in the Hess case (in the first of these cases), and the orthogonality of the vibration moment vector to the radius vector OG (in both cases), the body motion in these nonintegrable cases has a number of specific properties.
5.1. On the body motions in the general case of vibrations
As stated above, the Hamiltonian (3.27) of the reduced system for a body in the Hess case coincides with the Hamiltonian of a similar reduced system (with pv = 0) in the problem of dynamics of a Lagrange top with a vibrating suspension. This means that the motions of the axis of these two bodies (containing their centers of mass) are the same.
Hence, in particular, the equilibrium positions d = const, ^ = const of both reduced systems are the same, being stationary points of (the same) reduced potential energy. Sufficient conditions for their stability (with respect to perturbations of the quantities d, pi, p2), which are defined as conditions for the minimum of the potential energy, also coincide. However, in the Lagrange case, each equilibrium position of the reduced system corresponds to the stationary rotation of the body, while in the Hess case it corresponds to the family of equilibrium positions of the body, for which the angle p takes an arbitrary value. A similar property was noted in Section 4.1 when considering the integrable case.
The stationary rotations in the approximate problem of the dynamics of a Lagrange top with a vibrating suspension were investigated in detail in [26]. In connection with the foregoing, the results obtained in the article cited can be applied to describe the families of equilibrium positions of a body with the Hess mass geometry. We present the corresponding results, restricting ourselves to the case aXY = aXZ = aYZ = 0.
In this case, there are two families of equilibrium positions of the body, corresponding to the vertical position of the axis Oz', when the center of mass G is located above (d = 0) and below (d = n) of the suspension point. These equilibrium positions are stable (with respect to the above variables) when the condition
Bg
az =f —7 > max(a,Y, ay) mi
holds, and are unstable when changing the inequality signs to opposite ones. Here the upper and lower signs correspond to d = 0 and d = n.
There can also be families of equilibria for which the body axis is inclined to the vertical. They are described by the relations
with the quantities ai and a2 introduced by the formulae (3.17). The conditions for the existence of these families are given in parentheses. If, for definiteness, aX > aY (ai > a2), then the first one is stable for ai > Bg/(mi) and unstable for ai < -Bg/(mi); for this family, the axis of the
body is located in the OXZ plane. The second equilibrium family, for which the body axis lies in the plane OYZ, is unstable in the existence region.
For other vibration options considered in [26], the results obtained can be similarly applied to the problem studied in this paper.
5.2. The vibration symmetry case with nonequality in relation (3.28)
If the vibration parameters satisfy conditions (4.1), then for any body mass geometry, the vector mv of the vibration moment is always horizontal as well as the vector mg of the gravitational moment. For the Hess case at hand, these two vectors are collinear, since relation (3.3), after performing some transformations, is represented as
n,.= '\,jl(OGT)(Qg,<7).
B
We confine ourselves to considering the question of the existence of the particular body motions, being its stationary rotations around the axes fixed in the body and in the coordinate system OXYZ. They are analogs of Staude's permanent rotations in the dynamics of a heavy rigid body with a fixed point (see [33], as well as the monograph [34] and references therein).
For these motions, u = const, and Eq. (2.1) takes the form
u x Iu = mg + mv. (5.1)
From this relation it follows that the axis of permanent rotation of a body with any mass geometry can only be vertical if conditions (4.1) hold, and, therefore, u = wy (w = const). The permissible axes of permanent rotations lie on the surface of an elliptic cone in the space of the quantities 71, 72, 73 (projections of the vector 7 onto the principal inertia axes Ox, Oy, Oz), which is an analogue of Staude's cone for a body with a fixed point. The equation of this cone is obtained by scalar multiplication of both parts of Eq. (5.1) by the vector OG, which gives (taking into account the equality mg ■ OG = 0)
w2(y x I7) ■ OG = mv ■ OG. (5.2)
For a body with an arbitrary mass geometry and a body with a center of mass in the principal plane of inertia (but not satisfying the Hess conditions), the vectors mv and OG are not orthogonal, therefore Eq. (5.2) depends both on the angular velocity w of permanent rotation, and on the vibration parameters. For example, for the second of these cases (when yG = 0), this equation, written in terms of the projections of vectors in the system of principal axes of inertia, has the form
w272[(B - C)xg73 + (A - b)zg7I] =
= -m2{a^b~cax)j2(xgj3 - zgldm - b)cz2g - A(B - c)x2g]. (5'3)
This and the general cases are very difficult to study, since even the description of the set of permissible axes of permanent rotations is a very cumbersome problem.
If the body satisfies the Hess conditions, then the right-hand sides of Eqs. (5.2) and (5.3) vanish (due to the property mv ■ OG = 0, proven in Section 2), and these equations coincide exactly with the analogous equations for a body with a fixed point:
(7 x I7) ■ OG = 0 or 72 [(B - C)xg73 + (A - B)zg7i] = 0.
This equation describes a set of two planes
72 = 0 and (B - C)xg73 + (A - B)zg7i = 0 fixed in the body, in which the axes of the body permanent rotation can lie.
In the case 72 = 0 these axes lie in the plane Oxz containing the center of mass of the body. The equation of the second plane, with regard to the second condition in (2.6), can be rewritten as AxG71 + CzG73 = 0. Hence, due to the relations p = W71, r = W73, and yG = 0, we obtain equality (2.7). Thus, rotations around the axes from the second plane are realized in the integrable case, they were obtained in a different way in Section 4.2.
For the case 72 = 0, we project Eq. (5.1) onto the principal inertia axes Ox, Oy, Oz. The first and third equations are satisfied identically, and the second one has the form
(A - C)7i73W2 = -mg(zcY\ - xg73)
a
1 + - [Xoll + ZgJÍ)
From here we obtain the relation describing the set of admissible axes of rotations, it is the condition for the positivity of the value w2 in this equation. Since A > C, we have
7i73(zg7i - xg73)
a
1 + - [xoll + ZGji)
< 0.
Further research is not carried out here, it is an independent problem. Note that in the case considered in this section it is more convenient to choose the system of principal axes of inertia; the advantage in using the system of nonprincipal axes is lost.
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