On a Class of Precessions of a Rigid Body with a Fixed Point under the Action of Forces of Three Homogeneous Force Fields Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 2, pp. 249-264. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230604

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 70E05, 70E17, 70E55

On a Class of Precessions of a Rigid Body with a Fixed Point under the Action of Forces of Three Homogeneous Force Fields

G. V. Gorr

This paper is concerned with a special class of precessions of a rigid body having a fixed point in a force field which is a superposition of three homogeneous force fields. It is assumed that the velocity of proper rotation of the body is twice as large as its velocity of precession. The conditions for the existence of the precessions under study are written in the form of a system of algebraic equations for the parameters of the problem. Its solvability is proved for a dynamically symmetric body. It is proved that, if the ellipsoid of inertia of the body is a sphere, then the nutation angle is equal to arccos The resulting solution of the equations of motion of the body is represented as elliptic Jacobi functions.

Keywords: three homogeneous force fields, precessions, dynamically symmetric bodies, elliptic functions

Introduction

In the mathematical modeling of problems in rigid body dynamics and in the dynamics of systems of coupled rigid bodies, a crucial role is played by a class of motions that are called precessions. They are characterized by constancy of the angle between two axes 11 and l2 l is invariably related to the body and l2 is fixed in space). The application of precessions in the theory of gyroscopic devices is described in [1]. In the classical problem of the motion of a heavy rigid body, the following types of precessions are known: regular precessions of the Lagrange gyroscope [2] about the vertical; regular precessions of Grioli's gyroscope [3] about the inclined axis; Bressan's precessions of the Hess gyroscope about the horizontal axis [4]; precessions of the general type in Dokshevich's solution [5]. An overview of results on this problem is presented

Received May 20, 2023 Accepted June 22, 2023

This work was supported by the Russian Science Foundation under grant no. 19-71-30012.

Gennadii V. Gorr gvgorr@gmail.com

Steklov Mathematical Institute of Russian Academy of Science ul. Gubkina 8, Moscow, 119991 Russia

in [6, 7]. Numerous classes of precessions [7-9] have been obtained in generalized problems of the motion of a gyrostat with constant and variable gyrostatic momentum. Mention should also be made of results on the dynamics of an asymmetric liquid-filled rigid body which are devoted to the study of regular [10, 11] and semiregular precessions [12]. The problem of the motion of a system of coupled rigid bodies differs from those mentioned above in the order of the differential equations (it depends on the number of bodies in the system). Therefore, in deriving equations of motion of a system of coupled rigid bodies P. V. Kharlamov [13] considered only regular precessions of the system. Despite this circumstance, a solution to the equations of motion of the system of rigid bodies consisting of the Lagrange and Hess gyroscopes was obtained in [14] (the Lagrange gyroscopes perform regular precessions and the Hess gyroscopes perform semiregular precessions of the first type).

In the study of problems of the motion of a rigid body in a force field which is a superposition of two or three homogeneous force fields, some results were obtained on the integration of the equations of motion of Kovalevskaya's gyroscope [15-17] and on the bifurcation diagrams of first integrals [18]. Regular precessions of a rigid body in a triple homogeneous force field were studied in [19-21]. Thus, precessions of the general type in this problem were not considered.

This paper investigates conditions for the existence of a special class of precessions of the general type which is characterized by the equality p = 2tp, where p is the velocity of proper rotation of the body and tp is the velocity of its precession (p = const). These motions can also be interpreted as resonant precessions of the body. The conditions for the existence of solutions to the equations of motion are written as an algebraic system for the parameters of the problem. Its solvability is proved for a symmetric gyroscope rotating about the principal axis in the body which coincides with the axis l1. It is established that in the general case a0 G (0, xj, where = ~'~i¿f^' an<^ case where the ellipsoid of inertia of the body is a sphere,

a0 = where a0 is the cosine of the nutation angle. We note that, for a spherical gyroscope moving under the action of potential and gyroscopic forces, the following precession [22] takes place: p = tp, a0 = —

The new solution constructed in this paper to the problem of the motion of a body in a triple field is described by the elliptic Jacobi functions.

1. Formulation of the problem

Consider the motion of a rigid body having a fixed point in a force field which is a superposition of three homogeneous and constant force fields. Denote by Y, 7(1), 7(2) the unit vectors characterizing the directions of the forces P, P1, P2 of each of the fields, by C, C1, C2 the centers of reduction of the forces, let s = POC, r = P1OC1, p = P2OC2, denote by Oxyz a moving coordinate system, and let O be a fixed point. Suppose that the tensor of inertia of the body in the frame Oxyz has the value A = \\A-W (i, j = 1, 3). Assume that the body rotates about point O with the angular velocity u = w1i1 + w2i2 + w3i3 (i1, i2, i3 being the unit vectors of the frame Oxyz). For the vectors s, r, p we write the equations

s = S1i1 + S2i2 + Sзiз, r = T1i1 + r2i2 + ^^ P = P1i1 + P2i2 + P3i3• (L1)

Then the equations of motion of the body can be represented as (in [20, 21] they are written in a different form):

Aiu = Au x u + s x y + r x y(1) + P x Y(2), (1.2)

Y = Y x u, Y(1) = Y(1) x u, Y(2) = Y(2) x u, (1.3)

where the point above the variables u, y, 7(1), 7(2) denotes differentiation with respect to time t. In the formulae (1.2) and (1.3) we assume

Y • Y(1) =0, y(2) = Y x Y(1), IyI = 1» Y(1) = 1» (1.4)

i.e., the directions of the force fields will be characterized by the triple y, Y(j) (j = 1, 2). Then the equalities P = Py, Pi = PiY(i) (i = 1, 2) are obvious.

Consider the precessions of the body about the vector y. They are characterized by the invariant relation [6, 7]

a • y = a0 (a0 = cos d0), (1.5)

where d0 is the angle between the vectors a and y (a = 0, |a| = 1). The body's angular velocity vector satisfying the invariant relation (1.5) can be represented as follows [7]:

u = p a + ipY. (1.6)

The variables p, ^ and the constant d0 can be treated as Euler angles. Using the method [7], we write the value of the vector y(1) as

Y(1) = bo [aoY sin(^ + %) - a sin(^ + $0) + (a x Y) cosO + )]» (1.7)

where b0 = (a'0 = sin0o) and 0O is a constant.

We find the value of the vector y(2) from the second of equations (1.4):

Y(2) = b0 [a cos(^ + ^0) - a0Y cos(^ + ) + (a x y) sin(^ + ^0)j. (1.8)

Thus, in obtaining (1.7) and (1.8) it was assumed that a x y = 0, i.e., we exclude the case of uniform rotations of the body from consideration. Choose the moving coordinate system as follows: direct the vector i3 along the vector a. Then, using the invariant relation (1.5) and the first of equations (1.3), we obtain [6, 7]

Y = a0 sinp • i1 + a0 cos p • i2 + a0 • i3 (i3 = a). (1.9)

Using (1.6) and (1.9), we write the components w1, w2, w3 of the vector u as follows:

u1 = a0 ip sin p, w2 = a'0ipcos p, w3 = ^p + a0ip. (1.10)

Figure 1 presents a geometric interpretation of the body's precessions about the vector y (O(n( is a fixed coordinate system).

Remark 1. To describe the kinematic properties by relations (1.5)-(1.10), we use the method [7], which differs from the methods employed in [10, 13, 21, 22].

Remark 2. Equations (1.2) and (1.3) have the energy integral

Ay • Y - 2 (s • Y + r • Y(1) + p • 7(2)) = 2E, (1.11)

where E is a constant. As shown in [6, 7], the process of finding conditions for the existence of precessions in the problems of rigid body dynamics using (1.11) simplifies considerably.

2. Transformation of equation (1.2) taking into account the invariant relation (1.5)

Let us introduce the value of u from (1.6) into Eq. (1.2) and consider the resulting equation in the basis a, 7, a x 7 taking into account (1.7) and (1.8):

p(Aa • a) + p(Aa • 7) - tp2 [a • (Ay x y)] - [a • (s x 7)] —

- b0 Sin(p + p0) •{flQ[a • (r x 7) — a • p] + p • 7}— (2.1)

— b0 cos(p + po) •{r • y — a0 [(r • a) + a • (p x 7)]} = 0, p(Aa • y) + P(Ay • y) + 2pp[a • (Ay x 7)] + p2[7 • (a x Aa)] —

— bo sin(p + to) • {ao(P • 7) + [a • (r x 7) — (a • p)]}— (2.2)

— bo cos(p + po) • {ao(r • 7) — (a • r) + [a • (7 x p)]} = 0, <p[Aa(7 x a)] + P[Ay • (a x 7)] + pp[2(A7 • 7) — ^2Sp(A) — 2ao(Aa • 7)] +

+ p2[(Aa • y) — ao(Aa • a) + p2[ao(Ay • 7) — (Aa • 7)]— (2.3)

— (a • s) + ao(s • 7) — ao [(p • 7) cos(p + po) — (r • 7) sin(p + po)] = 0,

where Sp(A) = A11 + A22 + A33 is the trace of matrix A.

By analogy with (2.1)-(2.3) we write explicitly the integral (1.11) taking into account (1.5) and (1.6):

(Aa • a)p2 + 2(Aa • 7)pp + (Ay • 7)pt2 — 2 {(s • 7) + bo [sin(p + to) • (a«(r • 7) —

— (r • a) — p • (7 x a)) + cos(p + po) • (a • p — ao(p • 7) + r • (a x 7))]} = 2E. (2.4)

Let us introduce the notation

fo(p) = a0(«1 sin p + S2 cos p) + a0 S3 fo(p) = a0(s2 sinp - S1 cos p),

f1(p) = a0 [(a0r1 + P2) sin p + (a0T2 - P1) cos p - a0r3] ,

f2(p) = a0 [(t2 - a0P1) sin p - (a0P2 + r1) cos p + a0P3] ,

f3(p) = a0 [(P1 - a0T2) sin p + p + ao^) cos p], (2.5)

f4(p) = a0 [(T1 + a0P2) sin p + T - a^) cos p],

f5(p) = a0 [a0(s1 sin p + S2 cos p) - a0S^ , f6(p) = -a0 [00(t1 sin p + T2 cos p) + a0T3] ,

fz(p) = a0 [a0(P1 sin p + P2 cos p) + a^] . We first write the integral (2.4) by virtue of (2.5):

(Aa • a)p?2 + 2(Aa • y)ppip + (Ay • Y2-

- 2 [f0(p) + b0 (f1(p) sin(^ + ^0) + f2(p) cos(^ + ^0))] = 2E. (2.6)

Then we turn to Eqs. (2.1)-(2.3). By virtue of (2.5) we have

p(Aa • a) + Vi(Aa • y) - $2[a • (Ay x Y)] + f0(p)-

- b0 [f3(p) sin($ + $0) + f4(p) cos($ + $0)] = 0,

<p(Aa • y) + 'ip(AY • Y) - 2pip [a • (y x ay)] -

- b0 [f1(p) cos(p + p0) - f2(p) sin(p + p0)] = ^ (p [a • (y x Aa)] + p [a • (y x ay)] + pip 2(ay • Y) - a02Sp(A) - 2a0(Aa • y) +

+ p2 [(Aa • y) - a0(Aa • a)] + i2 K(Ay • Y) - (Aa • y)] + (2.9)

+ f5(p) + f6(p) sin(p + p0) + f7(p) cos(p + p0) = 0.

(2.7)

(2.8)

The representation of (2.1)-(2.4) in the form (2.6)-(2.9) is related to the solution of the problem of replacing one of Eqs. (2.7) and (2.8) with the integral (2.6). We calculate the time derivative from the left-hand side of Eq. (2.6) using the relations

Y = p(Y x ^ f0 = -P7o(P), f1(p) = pjf3(p), ff2(p) = pPf4(p).

Assuming that none of Eqs. (2.7) and (2.8) degenerates, we obtain an identity. That is, in this case the energy integral can be considered instead of one of Eqs. (2.7) and (2.8).

3. Description of resonant precessional motions of a rigid body

In equalities (2.5) and Eqs. (2.6)-(2.9) we set

p = 2p. (3.1)

By virtue of (3.1) such precessional motions may be classified with resonant precessions of the body. For further transformations we choose p as an auxiliary variable and write (2.5) as

fo(p) = ao(s1 sin2p + s2 cos2p) + aos3, fo(p) = ao(S2 sin 2p — si cos 2p), f1(P) = b1 sin2p + b1 cos 2p + bo, f2(p) = C1 sin 2p + c2 cos 2p + co,

f3(p) = —b2 sin 2p + b1 cos 2p, (3.2)

f4(p) = —c2 sin2p + c1 cos 2p,

h(p) = ao [ao(s1 sin2p + S2 cos2p) — aos^,

fe(p) = —ao [ao(^1 sin 2p + r2 cos 2p) + aor3],

f7(p) = ao [a'o(P1 sin 2p + P2 cos 2p) + a^],

where

bi = ao(aori + P2), b2 = ao(aor2 - Pi)> bo = -ao2r3) Ci = aO(r2 - aoPi), C2 = -a0(aoP2 + ri), Co = a'o2p:i.

(3.3)

To analyze Eqs. (2.6)-(2.8) we will use the functions $¿(0) (i, j = 1, 3); first, we present ^(p) and $2(P):

$i(P) = fi(P) sin(p + Po) + ¡2(^) COs(p + Po) =

= H3 sin 3p + G3 cos 3p + Hi sin p + Gi cos p, $2(P) = f3(P) sin(P + Po) + f4(P) COs(p + Po) =

= -G3 sin 3P + H3 cos 3P — Gli sin P + Hi cos P,

where

H-i = \ [(&2 + Ci)cosp0 + (&i -c2)sinp0], G3 = i [(fe2 + c1)sinp0 - (&! c2) cos po], (3.5)

Hi = H + Hi, Gi = Gi + Gi. (3.6)

Here

Hi = 2 Kci -&2)cos p0 + (c2 + 61)sin p0],

Gl = ^[(c2 +61)cosp0 - {C\ - 62)sinp0], (3J)

H = -ao2(r3 cos Po + P3 sin Po), Gi = a'o2(p3 cos Po - r3 sin Po).

Since the parameters Hi and G\ are expressed in terms of the original parameters, it is worthwhile to write the parameters Hi and Gi as

Hi = (1 - ao)[mi + m2(1 + ao)], Gi = (1 - ^[¡i + ¡2(1 + ao)], (3.8)

where

™>1 = y t(r2 +Pl) COS 00 - (P2 ~ r 1) Sin po], a!

h = -f[(P'2 ~ ^i)cos p0 + (p1+r2) sinpo],

(3.9)

m2 = — (r3 cos po + P3 sin po), ¡2 = P3 cos po — r3 sin po. RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2023, 19(2), 249 264

The relation between the parameters H£, G\ and m2, l2 is obvious:

Hi = a02m2, G\ = a02l2. (3.10) To investigate Eq. (2.9), it is also necessary to use the function

$3(—) = fa(—) sin(— + —0) + fz(—) cos(— + —0) =

= N3 sin3— + K3 cos 3— + N1 sin — + K1 cos (3.11)

where

N3 = -a0ni, K3 = -a0n2, N1 = a0(mi + aom2), Ki = a0(li + ao I2). (3.12) In (3.12), the parameters n1 and ^2 are as follows:

«1 = ^ao[(r2 ~~ Pi) cos 00 + (P2 +^l)sm '0o]) «2 = ^ao[(r2 - Pi) Sm 00 - (P2 + t\) COS t/)0].

(3.13)

(3.16)

Using the notation introduced in (3.3), (3.5) and (3.13), we have

H3 = (ao + 1)ni, G2 = (ao + 1)n2. (3.14)

Consider the integral (2.6). Taking into account (1.9), (3.1) and (3.4) in this integral, we obtain the differential equation

Fi (—)—2 = 2R(—), (3.15)

where

Fi (—) = L4 cos 4— + M4 sin 4— + L2 cos 2— + M2 sin 2— + L0, R(—) = R3 sin 3— + P3 cos 3— + R2 sin 2— + P2 cos 2— + R1 sin — + P1 cos — + R0. The following notation has been used in the formulae (3.16):

¿4 = ^o2(^22 - ¿11), M4 = a'Q2AV2, L2 = 2a'0(a0 + 2)A13,

M2 = 2 a'0(a0 + 2)A23, L0 = ¿a'02(An + A22) + A33(l + 2a0)2, (3_17)

R3 = b0H3, P3 = R2 = ^^ P2 = a'os2,

Ri = &0H1, Pi = ^Gi, R0 = a0S3 + E. We write Eq. (2.7) using the formulae (1.9), (3.1) and (3.4):

F2(—)—= F3 (—)—2 + $4(—), (3.18)

where

F2(—) = K2 sin 2— + N2 cos 2— + K0,

F3(—) = M4 cos 4— - L4 sin 4— + M2 cos 2— - L2 sin 2—, (3.19)

$4(—) = —P3 sin 3— + R3 cos 3— — P2 sin 2— + R2 cos 2— — b0Gi sin — + b0H1 cos Here, the following values are taken as the new parameters K2, N2, K0, M2, L2:

K2 = a0 A13, N2 = a0 a0 A13, K0 = (a0 + 2)A33, M2 = a0a0 L2 = a0 a0A23. (3.20)

Thus, in the first step of analysis of the system (2.6)-(2.8) we have obtained two equations: (3.15) and (3.18) (this is due to the fact that, as noted above, all Eqs. (2.6)-(2.8) are dependent).

4. Analysis of consistency of equations (3.15) and (3.18)

Differentiating both sides of Eq. (3.15) with respect to time and substituting p from (3.18) into the resulting equation (F2(p) = 0), we establish an equation that contains the function p2. Taking into account p2 from (3.15) in this equation, we obtain

R(p) [2F1(p)F3(p)+ F2(p)F1 (p)] + F1 (p) [F1 (p)$4(p) — F2(pR (p)] =0. (4.1)

The condition of consistency of Eqs. (3.15) and (3.20) is that, when the functions (3.16) and (3.19) are substituted into equality (4.1), it must be an identity in the variable p. As a result, we obtain a system of algebraic equations for the parameters (3.17) and (3.19), which depend on the parameters of the problem. We study the conditions which follow from the fact that the coefficients with sin 11p, cos 11p are equal to zero:

M4R3 — L4 P3 = 0, M4P3 + L4R3 = 0. (4.2)

Assuming P| + R2 =0 in (4.2), we obtain

M4 = 0, L4 = 0. (4.3)

Using the notation (3.17), we find conditions for A- (i, j = 1, 3):

A12 = 0, A22 = An. (4.4)

If conditions (4.4) are satisfied, the functions F1(p) and F3(p) simplify, which allows us to consider the equalities in which the coefficients with cos 7p, sin 7p are equal to zero:

RN + P3K2 = 0, R3K2 - P3N2 = 0,

which, by virtue of the assumption P3 = 0, R3 = 0, imply N2 = 0, K2 = 0, or, using the notation (3.20), yield

A23 = 0, A13 = 0. (4.5)

We write F1(p), F2(p), F3(p) taking into account conditions (4.4), (4.5) and the notation (3.17), (3.19), (3.20):

F1 = Lo = a'02An + (1 + 2ao)2 A33, F2 = K0 = (aQ + 2)A33, F:i(p)=0. (4.6)

Then, by virtue of (4.6), from Eq. (4.1) we obtain the last condition (again we assume H3 = 0, G3 =0):

(ao + 1)An - (4ao + 5)A33 = 0. (4.7)

Consequently, under the assumption P3 = 0, R3 = 0, H3 = 0, G3 = 0, Eqs. (3.15) and (3.18) have been studied. To obtain the other conditions for the existence of the invariant relation: p = 2p, it is necessary to make use of equation (2.9).

5. Analysis of equation (2.9)

Let us introduce the notation

F4(—) = L4 sin 4— — M4 cos 4— + L2 sin 2— — M2* cos 2—, (5.1)

F5(—) = (a0 + 4)(L4 cos 4— + M4 sin 4—) + M2 sin 2— + L2 cos 2— + M0, (5.2)

F6(—) = N3 sin3— + K3 cos 3— + a0a0(s1 sin 2— + s2 cos 2—) + N1 sin — + K1 cos — — a'02s3,

(5.3)

where the new parameters L2,, M|, L2, M2, M0 are

Li = a0 (a0 + 2)^23, M2* = a0(a0 + 2)AW, L2 = a0(2a0 + 4a0 + 3)^3, M = a0 (2a2 + 4a0 + 3)A^,

2

ci

Mo = ^-MAi + ^22) - 2(ao + 2)A33].

(5.4)

Using (5.1)-(5.3), we write Eq. (2.9) as

F4(—)— = —F5M—2 — F6(—). (5.5)

Applying the method of Section 4 to analysis of Eqs. (3.15) and (5.5), we obtain the equation R(—) [2Fi(—)F5(—) — F4(—)F1 (—)] + Fi(—) [Fi(—)F6(—) + F4(—R(—)] = 0. (5.6)

In Section 4 it is shown that, when H3 = 0 and G3 = 0, conditions (4.4), (4.5) and (4.7) are satisfied. Therefore, we first consider equation (5.6) in the presence of these conditions. By virtue of the notation (5.4), we obtain F4(—) = 0, F5(—) = M0 from (5.1) and (5.2). We write Eq. (5.6) in the following form (taking (3.12) and (3.16) into account):

2b0M0[(a0 + 1)(n1 sin3— + n2 cos 3—) + ...] — L0[a0(n1 sin3— + n2 cos 3—) + ...] = 0. (5.7)

Here the ellipsis denotes terms containing the functions sin 2—, cos 2—, sin —, cos —. Since it follows from (3.14) with n1 = 0, n2 = 0 that H3 = 0, G3 = 0, we obtain from (5.7)

2b0M0(a0 + 1) — a0 [a02An + (1 + 2a0^33] = 0, (5.8)

where by virtue of (5.4)

M0 = a02 KA11 — (a0 + 2)^33]. (5.9) Substituting M0 from (5.9) into (5.8) and using (4.7), we obtain the condition

6a02 + a0 — 10 = 0. (5.10)

It is obvious that the roots of equation (5.10)

00 =-12-

do not belong to the interval (—1; 1). Therefore, we need to set in (5.7) n1 = 0, n2 = 0 or

H3 = 0, G3 = 0. (5.11)

Consider the equalities n1 =0 and n2 = 0 taking into account the formulae (3.13). Then we find conditions for the parameters of the force fields

P1 = r2, P2 = —r1. (5.12)

Let us formulate the result obtained in Sections 4 and 5. Equations (5.11) and (5.12) are a necessary condition for the existence of the invariant relation (3.1) of Eqs. (2.6)-(2.9).

6. The case H3 = 0, G3 = 0

We write the functions (3.4), (3.11) and $4(p) from (3.19) and R(p) from the system (3.16) under conditions (5.11) and (5.12):

$1(p) = H1 sin p + G1 cos p, $2(p) = —G1 sin p + H1 cos p,

$3(p) = N1 sin p + K1 cos p,

11 (6.1) R(p) = R2 sin 2p + P2 cos 2p + R1 sinp + P1 cos p + R0,

$4(p) = —P2 sin 2p + R2 cos 2p — b0G1 sin p + b0H1 cos p,

where the coefficients of the functions (6.1) are defined by Eqs. (3.8), (3.12) and by the relations from system (3.17). By virtue of (5.12) the values of (3.9) simplify to

m1 = a0(r2 cos p0 + r1 sin p0), l1 = a0(r2 sin p0 — r1 cos p0). (6.2)

The parameters H1 and G1 possess a similar property:

il1 = (1 — a0 )m1, G1 = (1 — a0)l1. (6.3)

We establish the relation of the parameters H1, G1 to the parameters H1 and G1 using the formulae (3.8) and (6.3):

H1 = + a"2m2 = (1 — a0)[m1 + (1 + a0)m2], G1 = G1 + a0l2 = (1 — 00)^1 + (1 + ^0)12], (6.4)

where m2 and l2 are written in (3.9). Consider equality (4.1), in which R(p), $4(p) have the form presented in (6.1), and the parameters L4 and M4, which appear in the functions F1(p) and F3(p), are different from zero. The requirement that equality (4.1) be an identity in p leads to the conditions

s1 = 0, s2 = 0. (6.5)

A repeated study of Eq. (4.1) makes it only possible to obtain the conditions

M4 + - L4 (G, + = 0, M4 (g! + ^ + M4 (H! + = 0.

Under the assumption M4 = 0, L4 = 0, the following equalities follow from the system (6.6):

H1 + ±H1 = 0, Gl + l-Gl= 0,

from which, by virtue of (6.2)-(6.4), we find additional restrictions on the parameters of the problem:

3ri(1 - Oq) - 2aQp3 = 0, 3^(1 - ao) - 2a'0r3 = 0. (6.7)

If we assume = 0 (i = 1, 3) in (6.7), then from (5.12) we have pi = 0 (i = 1, 3). That is to say, the body is acted upon only by forces of one force field, which is of no interest. Further inspection of Eq. (4.1) and Eq. (5.6) under conditions (6.5), (6.7), M4 = 0, L4 = 0 leads to lengthy calculations and is only advisable in proving theorems on the nonexistence of a solution. Therefore, a more important circumstance is obtaining a concrete solution to Eqs. (2.6)-(2.9).

7. A new solution of equations (2.6)-(2.9)

Let us examine conditions for the existence of a solution to Eqs. (2.6)-(2.9) in the case of dynamical symmetry relative to the axis passing through the vector a. Suppose that Eqs. (4.4) and (4.5) are satisfied, and assume at the start that s1 = 0, s2 = 0. Then for the functions Fi(—) (i = 1, 3), Eqs. (4.6) hold, and for the functions F4(tp), F5(tp) the expressions are as follows: F4(—) = 0, F5(—) = a02[a0An — a + 2)^33].

Suppose that in Eqs. (4.1), (4.6) s1 = 0, s2 = 0. Using in these equations the values of Fi, F5, F6, R(—) from (3.16) (with (4.4), (4.5) and H3 = 0, G3 = 0), from Eqs. (4.1) and (5.6) we obtain

(L0 — 2K0)si = 0, (2M0 + a0L0>i = 0 (i = 1, 2). (7.1)

If we assume in (7.1) that s1 =0, s2 = 0, then we obtain the equations

L0 — 2K0 = 0, 2M0 + a0L0 = 0. (7.2)

Let us substitute L0 and K0 from (4.6) into Eqs. (7.2):

a02Ai + a0(a0 + 2) A3 = 0, (7.3)

3a02 a0A1 + (a0 + 2)(3a0 + 2a0 — 2)A3 = 0. (7.4)

It is obvious that Eqs. (7.3) and (7.4) cannot be satisfied simultaneously. Therefore, from (7.1) we obtain the conditions s2 = 0, s1 = 0.

Let us write the other conditions which follow from the requirement that Eqs. (5.6) be identities in — in the presence of Eqs. (4.4), (4.6), F4(—) = 0, F5(—) = M0, and take into account in F6(—) the equations N3 = 0, K3 = 0, and in R(—) the equations P3 = 0, R3 = 0. Then we obtain the following system of algebraic equations:

L0H1 — K0H1 = 0 L0G1 — K0G1 = 0 rx

(7.5)

2M0 b0H1 + L0N1 = 0, 2M0b0 G1 + L0K1 = 0,

where

Hi = Hi + (^m^ Gi = (5i + a02 ¿2, Hi = (1 — a0) m 1, Gi = (1 — a0 )li, (76) N1 = a0(mi + a0m-2), Ki = a0(li + a0^).

Let us substitute the values of H1, H1, G1, (G1, N1, K1 from (7.6) into the system (7.5) and represent it in the form

(L0 — K0)m1 — (1 + a0)K0m2 = 0, (7 7) [2M0 + (1 + a0)L0 ]mi + (1 + a0)(2M0 + L0a0)m2 = 0, (L0 — K0 )li — (1 + a0)K0 ¿2 = 0, [2M0 + (1 + a0)L0 ]li + (1 + a0)(2M0 + L0a0 % = 0. (7.8)

If we assume 2M0 + L0a0 = 0 in Eqs. (7.7) and (7.8), then we find the equations m1 = 0, m2 = 0, l1 = 0, l2 = 0. If these equations are satisfied, from (3.9) and (5.12) we obtain pi = 0, fj = 0, where (i = 1, 3). It follows from Eq. (1.2) that the body is acted upon only by the forces of one force field. By virtue of the equations s2 = 0, s1 = 0, the force P is applied at the end of the vector s. This case is of no interest due to the problem statement formulated above. It is considered in [6].

Let 2M0 + a0L0 = 0 in (7.7) and (7.8). Using the conditions m1 = 0, m2 = 0, l1 = 0, l2 = 0, from (7.7) we obtain 2M0 + a0L0 + K0 = 0. Hence, taking into account the values of the parameters M0 from (5.4) and L0 and K0 from (4.6), we establish the condition

3a0(1 - 2a0)A11 + 2(3a2 + a0 - 1)A33 = 0. (7.9)

If we set A33 = A11 in equation (7.9) (by virtue of A22 = A11 the ellipsoid of inertia of the body is a sphere), then from (7.9) we have (a0 + 2)(3a0 — 1) = 0. Since a0 + 2 > 0, we establish the following value of a0:

ao = I- (7-10)

Since it was found above that s2 = 0 and s1 = 0, the function p(t) from (3.15) satisfies the differential equation

02 = ~riF M1 " a-o)Lo(™<i sinp + h cos 0) + K0(a0s3 + E)}. (7.11)

L0K0

If condition (7.9) is satisfied, it suffices to consider in the system of equations (7.7) and (7.8) only the first two equations from (7.7) and (7.8), respectively. Using the values of m1, l1 from (6.2) and the values of m2, l2 from (3.9), we write them as

¡1 cos p0 + ¡2 sin p0 = 0, ¡2 cos p0 — ¡1 sin p0 = 0, (7.12)

where

¡1 = a0 (L0 — K0)r2 + (1 + a0)K0r3, ¡2 = a0(L0 — K0 )r1 + (1+ a0 )K0P3. (7.13)

It follows from the system of equations (7.12) that ¡1 =0, ¡2 = 0, or, in explicit form, that is, taking into account the values of L0 from (3.17) with A22 = A11, K0 from (3.20), these equations are

a0r2 [(1 — 00)An + (4a0 — 1)A33 ] + K + 2)r3 A33 = 0,

aW(1 — a0)An + (4a0 — 1)A33 ] + K + 2)p3A33 = 0. .

In the analysis of Eq. (5.6) we wrote out only the conditions in which free terms were ignored. Consequently, it is necessary to additionally consider the equation

' 2 T

To study Eq. (7.9)

in the case A33 = An, we apply the graph-analytical method by setting y = x = a0. Then from (7.9) we find the function y(x):

By virtue of inequalities of the triangle for the principal moments of inertia A22 = A11, A11, A33, it follows that, y > Using this inequality, from (7.16) we find the condition x e (0, x*), where

18

Hence, condition (7.9) is satisfied if

ao e (0, x*), (7.18)

where x* is defined by the formula (7.17) and it is obvious that the value (7.10) belongs to the set (7.18).

8. A numerical example. Reduction of the solution to elliptic Jacobi functions

For illustrative purposes, we consider the problem of reducing the solution to elliptic Jacobi functions under the condition that A33 = A11 and that the parameter a0 has the value (7.10). We write the obvious equalities for the parameters

11 7 16

Lo = YAn' k° = 3Al1' M° =

(8.1)

Under the conditions a0 = | and (8.1) the function (7.11) satisfies the equation

02 = -J-\\ \(r2\/3-r^j simp- (tl \/3 + r2) cos 0 (8-2)

A 17

Denote

' 2

Then by virtue of (8.3) from (8.2) it follows that

siD/^o = / 2 ,2 2V cos/j0 = - 2 2 1 2V (8.3)

2 (r2 + r2) 2 (r2 + r2)

? = T

(8.4)

Let us introduce new parameters

88 y/rl + r\ 2 _ 4 sjr'l + v2

-1 % — -—

308 ' 7An

A0 = 2, ** = V;J, 2 < 1. (8-5)

kg

In the formulae (8.1) and (8.4) we assume s3 < 0, which does not restrict the generality of the problem. Make a change of variable 0:

0 = £ - Po, (8.6)

where £ is a new independent variable. Consequently, using the formulae (8.4)-(8.6), we obtain

é = J-^-Vl-x2 sin2t. (8.7)

V Aii

Write (8.7) in the integral form

de I X^

+o

ht-tq) = t. (8.8)

J \/l — X2 sin2 £ V

e0

From the formula (8.8) we have

e = am t, sin e(r) = sn(r), (8.9)

where e(T) and sn(T) are elliptic Jacobi functions. By virtue of (8.6) and (8.9) P(t) = am t — (30 and hence

sin P(t ) = cos (30 sn(T) — sin (30cn(T). (8.10)

Consider the construction of the solution y(t), u(t), Y(1)(t), Y(2)(t) for Eqs. (1.2) and (1.3). Since, by assumption (3.1), p = 2p, we define the dependence y(t) from (1.9):

Y(t) = a'0 sin 2p(T)i1 + a'0 cos 2p(T)i2 + a0i3, (8.11)

and the components u from (1.10):

u1 = a'0ipsin2p, w2 = a0 ipcos2p, w3 = i(a0 + 2), (8.12)

where a'0 = a0 = We obtain the vectors 7^(ip) (i = 1, 2) from the formulae (2.7) and (2.8), in which

11

(8.13)

and the vector 7(t) has the value (8.11).

Thus, the conditions for the existence of precessions of a dynamically symmetric body, for which Lp = 2ip, are the equations p1 = r2, p2 = —r1, s1 = 0, s2 = 0, (7.9). In the case of spherical

mass distribution, we replace Eq. (7.9) with condition (7.10), and in (7.14) we assume a0 = A33 = A11. The solution of the initial equations of motion of the body is expressed by elliptic Jacobi functions.

Conclusion

This paper treats the problem of the motion of a rigid body with a fixed point under the action of forces of three homogeneous force fields. It is assumed that the angle between two axes one of which is invariably attached to the body and the other is fixed in space is constant (this motion is called precession of the body). An analysis is made of a class of precessions of the body which is characterized by the condition Lp = 2p, where Lp is the velocity of proper rotation and ip is the velocity of precession of the body. A new solution of the equations of motion of a dynamically symmetric rigid body is established. It is shown that the center of reduction of forces of one force field belongs to the principal axis of the ellipsoid of inertia, and the projections of the vectors p and r onto the plane orthogonal to the axis of dynamical symmetry of the body are orthogonal as well. A condition for the principal moments of inertia and the nutation angle is obtained from which for the spherical mass distribution of the body it follows that the nutation angle is equal to arccos 4. The solution thus found is described by elliptic functions of time.

£

Conflict of interest

The author declare that he have no conflict of interest.

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