YEHIA, H. 31. : Particular Tntegrable Cases in Rigid Body Dynamics 33

Z.4M31. Z. angew. Math. Meoh. G S (1988) 1, 33 -37

YEHIA, H. M.

Particular Integrable Cases in Rigid Body Dynamics

E s wird das Problem der Bewegung eines festen Korpers um einen festen Punkt unter der Einwirkung axialsymmetrischer potential- und gyroskopischer Krafte betrachtet. Dabei werden losbare Spezialfiille gefunden, die die Ergebnisse von H F ss, K h a r l a m o v m d anderen verallgemeinern.

The problem of motion of a rigid body about a fixed point under the action of axisymmetric potential and gyroscopic forces is Considered. Particular solvable cases are found, which generalize the results due to H e s s , K h a r l a m o v ond other&.

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1. Introduction

I n our works [l], [ 2 ] , [3] we have considered the problem of motion of a rigid body about a fixed point under the action of' axisymmetric potential and gyroscopic forces. This problem can be modelled by the motion of a heavy, charged and magnetized gyrostat in a combination of axisymmetric gravitational, electric and magnetic fields, It includes as special versions most of the cases considered earlier in the dynamics of rigid bodies about a fixed point and also the problem of motion of an arbitrary rigid body in an ideal incompressible fluid.

I n 131 we have pointed out six general integrable cases of the above problem. In the present work we consider certain particular solvable cases which essentially generalize earlier results of several authors.

The equations of motion will be taken in the form of EULER-POISSON [2] i3V

ul t 0 x (a1 + p ) = y x -, p + w x y = 0 , (1) 3Y where I = diag ( A , B, G) is the inertia matrix of the body a t the fixed point, UJ = (p, q, r ) is the angular velocity of the body, y = (y l , r2, y3) is the unit vector along the common axis of symmetry of the system of forces applied to the hody. The potential V and the vector p which characterizes the gyroscopic forces are considered as fuiictions of y only. We can always express p in terms of two scalar functions in the form [2]:

T h e system (1) admits two general integrals f W ~ * W + Y = h , 01.y + F = f , (31, (4)

where h and f are arbitrary constants. We shall consider here two cases in which equations (1) admit one or three linear invariant relations.

2. Invariant relation of Hess type

We shall now find the form of V ( y ) and p ( y ) for which the equations (1) admit) an invariant relation of the form ,I = A cosap $- C sinocr + b = 0 , (3 1

We first note that we can consider V and b as functions of yl , y3 only, by virtue of the geometric integral. in which ct is a constant and b = b ( y ) .

Differentiating the expression ( 5 ) and using (1) we obtain

( A - B) sinRp + ( B - C) c o s w + sincyu, - coscupU, 4-

Thus, ( 5 ) will be an invariant relation for ( 1 ) whenever

3 % . aiigrw. Ilirth. Mech., Bd. 68, €1. I

34 ZAM/IICI. Z. angew. Math. Nech. 68 (1988) 1

Now we introduce new variables : B x = coscuy, + sin ay, , rj = -sill ayl + ~ o s a j ) ~ , 11 = -

I i l C ‘ ( A cos‘vyl + C sinixy,) . (10)

Note that these variables satisfy the relations B ( A - C) . s1n x cos ny , x z + l j z = y 1 2 2 + y 3 = 1 - y ; , y = x -

A C From (9) i t follows that the potential must have the form

where v is an arbitrary function of x. I n the remaining equations ( 7 ) ) (8) one can arbitrarily choose b and pl, in which case,uu, and,uu, will be uniquely determined. Hence, the invariant relation (5) exists in a wide class of problems characterized by three arbitrary functions b , v, pl.

We proceed now to fit the above inverse result to the application in specific problems with given p. When A # B # C, using ( 2 ) and ( lo ) , we can reduce equation (ti), in the independent Variables 5, y, to

a - (P - yb) = 0 . aY

Hence we obtain

P - v(x) b = ~~~~

!I

Using (7) , (13) and (2) we get where g is an arbitrary function.

The formulas (12) -(14) characterise the most general case of existence of the invariant relation (5). However, in the most significant physical cases the functions V,yc are continuous on the whole Poisson’s sphere. To avoid the singularity in V and ,uz on the plane y = 0, we must choose g(x) = F ( z , y)/7J=o = F ( x , 0).

I n the most general case, as in thc classical case of HESS 1.21, the variable z can be determined by a quadrature. I n fact, using the integrals.

Apz + Bq2 f Cr2 + 21’ L 2 h , .4pyl + Bqyz $- Cry, -/- F = f ,

together with the relation ( 6 ) we can obtain :

B(1 - Z2) 4 -7 i E + y,if - 8 ) , I-- B

cosxtl(J’ - g ) - -sill a ( 1 - r2) b , A

C(1 - xz) r = cosx V K 7.

where R = R ( x ) == 2B(1 - 5’) (h - V ) - (f - 9)”

(16)

From Poisson’s equations and using (15) we rcadily obtain the eqiiation :

(16) /- Bi = ~ R ( z ) ,

which determines x in terms of the time. This rariable x is the cosine of the angle between the line whose direction cosines in the body are cos a, 0, sin a and the vector y .

On the other hand, the determination of the rotational motion of the body about t>his line can be reduced to a first order nonlinear differential equation whose solution is extremely difficult.

As an example we now consider the case when the body mores under the action of potential forces only. I n this case p = 0, F = b = g = 0. The HESS invariant relation exists for any arbitrary choice of the potential

v = V ( Z ) . (17) The classical case of H ~ s s corresponds to the choice of w as a linear function in x.

Potentials of the type (17) can be realized by adding to the body, in the problem of HESS, a distribution of magnitization (or electric charges) with axial symmetry about the line joining the centre of mass with the fixed point and any external magnetif (or electric) field w i t h axial symmetry around the vcrticsl passing through the fixed point.

As a second example we choose

SIEIIIA, H. Rl. : Particular Integrable Cases in Rigid Body Dynamics 35

This result generalizes by an arbitrary function and several parameters all known results for different prob- lems of motion of rigid body. When a,, = aI2 = aZ2 = 0, v(z) is linear we have the case of SRETENSKY [ 5 ] of the motion of a heavy gyrostat. When v ( x ) is quadratic we obtain a case of motion of a multiconnected body in a liquid [3]. The same result can be interpreted as a case of motion of a charged rigid body under the influence of a super- position of parallel uniform gravitational and magnetic fields and an electric field with axial symmetry aronnd the rertical throngh the fixed point [2].

3. The ease of axial dyiiamical symmetry

When the body possesses axial dynamical symmetry, say, A = B, then cosoc = 0 and the variables x, 9 i n (10) are no longer independent. From equation (8) we obtain instead of (13) the relation :

1 Y 3

1) = --F(Y,, Y 3 ) - d Y 3 ) * (13')

The reinainiiig formulas all remain valid after setting B = A .

4. The case of three linear invariant relations

Sow, let the system (1) admit three invariant relations of the form

a = f a y ) > (18)

j , +n x y = o . (19)

and hence i t will reduce to the equations

Equations (19) always admit another integral independent of time in addition to the geometric integral. I n fact, from (19), we get the equation

f2. dy = 0 , for which we can always find an integrating factor B ( y ) (say) by Euler's theorem. This is because i t contains, by virtue of the geometric integral, two variables only. We denote this additional integral by

Y(y) = const. (20) This is equivalent to the assertion that R can always be represented in the form

where

tioris on the functions V and y. In fact, the energy integral gives

is a certain function of y . The condition that Euler's part of (1) is satisfied identically by

$aI . $2 + V = const., and hence we must take V in the form

v = -+&?I , f.2 + u p ) . , where v is an arbitrary function of 'P. 3'

(81)

the choice of w in (18) implies certain rwtrica-

(23)

36 ZAMM . Z. angew. Math. Mech. 68 (1988) 1

Taking also p as in ( 2 ) and using the cyclic integral, we deduce

P = - - y I . Q + S ( W 9 ( 2 3 ) where g is another arbitrary function.

Substitutingfrom ( 2 ) ) ( 2 2 ) and (23) into Eider's equations we obtain an identity under the sole restriction that

Therefore, we get for p the expression

The expressions ( 2 2 ) ) (24) characterise the most general case when (1) admits the invariant relations of the

If i t is difficult to get theintegral (20)) or if the function Y possesses undesired properties such as multivalued-

A s a concrete example we take the case

type (18).

ness and singularities, we may take the functions v('P) and g(Y) as constants or zeros.

Q = yik! + 111. ) (26) where M , m are a constant real matrix and a rector respectively. Consider first the case when iM is symmetric. Then we have

! P = + y M . y + m . y , & y ) = 1 , and hence

V = - ' ( 2 Y M + m) I . (YM + rn) + qw f

p = g'('.P) (yB + rn) - ? ( M I t I X ) - m I + [Tr M I - v'[!?')]y If we further choose

e(Y) = UP, g ( ! P ) = b y , where a and b are constants, we can write (28) as

where

- V = f 7 J . JP + S . y , tt = k +;, I ( ,

J = - M l M + aJrl + ed , 12 = m(b6 - I ) ,

s = m(ad - I J l ) , = b J 1 - M I - I X 4- (Tr ill1 - a ) d ,

(39)

E is an arbitrary constant and b is the unit matrix. Thus, we have obtained a case of the problem of motion of a multiconnected rigid body in a liquid [3], in which

18 free parameters are present. Adding three parameters, which occur during the solution of (19), we get a solution containing 21 parameters. If we return to Kirchhoff's orKharlemov's variables, we add three parameters correspond- ing to the nonsymmetry of the matrix coefficient of the mixed terms in the kinetic energy. The result which coil- tains 24 free parameters out of 33, is n generalization by 8 parameters of KHARLA4MOV's result [(i]. It also generalizes our previous result in [ 7 ] .

We note that all the variables in our case can, generally speeking, be expressed as elliptic functions of time. In fact, if det M # 0, we have

y = QAf-1 - mx-1 , and (19) can be written as

(SZN-1). + SZ x (SZM-l - m 3 1 - 1 ) = 0 , which formally coincides with the equations of motion of a gyrostat with the intertia matrix $1-1 and the gyrostatic moment - m M - l . The solution of the last problem has been obtained by VOLTRRRA [ 8 ] in terms of Weierstrass' sigma functions. A solutiorr in Jacobi's functiorts was obtained by WITTEXBURG 191.

Another special case of our result was obtained by K H A R L A M o v A [ 101 in her work on the problem of motion of a gyrostat in the approximate Sewtonian field of a point mass. This case corresponds to the choice

M = oc(bd - 2I)-1 , ci : Tr(N1) - c, , where LY is a constant.

p the same expressions as in (35)) where We now consider (Zti), when 111 is not synimetric. Taking v(Y) = g(Y) = 0 in (28) and (31)) we get for V and

J = - a M I M T , s = - m I * W , 12 = - m I , R = -(171I + I B I T ) . (30) AS in the case of symmetry of 111, this result can be interpreted as a case of motion of a rigid body in a liquid

are

The explicit solution of the problem corresponding to the choice (28) can be easily obtained in a certain special

~

or a s the motion of a charged gyrostat in a combination of classical fields. This is because the matrices J and symmetric. A s far as we know, no special cases of this result were obtained earlier.

case which we now consider.

TEHIA, H. M.: Particular Integrable Cases in Rigid Body Dynamics 37

Tn the most general system fixed in t h e body axes around the th i rd by

case, the mat r ix 31 has at least one real eigen-value. Taking the third axis of the Cart.esiari t o coincide with the eigen-vector corresponding to this value a n d rotating t h e o ther two a suitable angle, we can always render M to the form

If i n the same coordinate system M32 = wb3 = 0, the first pair of equations (19) takes the form 9 1 + y3[H12?1 + ( M 2 Z - M33) Y 2 + = 9 9 2 f Y 3 [ ( M 3 3 - Mil) 71 - M'21Y2 - = . (31)

Introducing a new variable z by the relation

i 32)

the variables y l , y 2 can easily be found in terms oft and hence y3 . The expressions for the variables in te rms of the time a re obtained through the inversion of (32).

References

1 YEHIA, H. M., New solutions, of the problem of motion of gyrostat in potential and magnetic fields, Vestn. Mosk. Univ., Ser. 1 ,

2 YEHIA, H. M., On the motion of a rigid body acted upon by potential and gyroscopic forces. I: The equations of motion and their

3 YEEIA, H. M., On the motion of a rigid body acted upon by potential and gyroscopic forces. 11: A new form of the equations of

4 HEM, W., Uber die Eulerschen Bewegungsgleichungen und uber eine neue particulare Losung des Problems der Bewegung eines

5 STRETINSKY, L. N., On some cases of motion of heavy rigid body with gyroscope, Vestn. Mosk. Univ., Math.-Mekh. 1963,

6 KHARLANIOV, P. V., On the solutions of the equations of rigid body dynamics, - Prikl. Mat. Mekh. 29 (1956) No. 3. 7 YEHIA, H. M., On certain class of motions of a gyrostat in a superposition of three classical fields, Vestn. Mosk. Univ., Ser. 1,

8 VOLTERRA, V., Sur la th6orie des variations des latitudes, Acta Math. 22 (1899). 9 WITTENBURQ, J., Beitrage zur Dynamik von Gyrostaten, Accad. Naz. dei Lincei, Quad., 217 (1975).

Mat.-Mekh. 1985, No. 5.

transformation, J. MBcan. th6or. appl. 6 (1986).

motion of a multiconneeted rigid body in an ideal incompressible fluid, J. MBcan. th6or. appl. 6 (1986).

starren schweren Korpers um einen festen Punkt, Math. Ann. 37 (1890) 2.

No. 3.

Mat.-Mekh. 1986, No. 1.

10 KHARLAMOVA, E. I., Certain solutions of the problem of motion of a body having a fixed point, Prikl. Mat. Mekh. 29 (1965) No. 4.

Received May 26, 1986

Address: Dr. H. M. YEHIA, Department of Applied Mathematics and Theoretical Physics, University of Liverpool, P.O. Box 147, Liverpool L 69 3 BX, Great Britain. Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt