The integral invariants of n gyrostats

MECHANICS RESEARCH COMMUNICATIONS Vol. 17 (2), 75-80, 1990. Printed in the USA. 0093-6413/90 $3.00 + .00 Copyright (c) 1990 Pergamon Press plc

THE INTEGRAL I N V A R I A N T S

OF n GYROSTATS

A. G. MAVRAGANIS

Department of Engineering Science Section of Mechanics National Technical University of Athens 5 Heroes of Polytechnion Ave., GR-15773,

ATHENS, GREECE

(Received 31 January 89; accep~d mr print 19 December 198~ INTRODUCTION Cid and Vigueras working on the problem of n isolated gyrostats have proved [I] that the motion is controlled by nine f i r s t integrals and, provided that the relative angular momenta and kinetic energy of the mobile parts of the system are time-independent, by one more integral which is theJacobi'sintegral. The present result generalizes an analogous conclusion of Duboshin's work [2], [ 3 ] , [ 4 ] , on the motion of n rigid bodies, whose elementary particles interact in accordance to Newton's law or more general laws. Since the n gyrostats are acted upon the internal forces only, W nicharesupposed proportional to the product of all masses obtained in pairs, and as functions of the respective distances, only the individual rotor-translatory motion ofthegyrostatsisconsidered, the long-time behaviour of the system as a whole being ignored. Based on these integrals we derive an equal number of integral-invariants, whose role in the study of such a system is, just like of any dynamical system, quite reasonable because they can improvethequalitative approach of i t s motion. Besides they may be used to check the accuracy of the numerical integration of the equations of motion when other c r i t e r i a fail or do not exist at all. We also give two other integral-invariantsofextremeorder, i.e. i and n, by particularizing into the case of question, the invariancy which characterizes the "circulation" of the system in the phase space and the volume occupied by all parts of i t . THE EOUATIONS OF MOTION Let ( g i ) ,

i : 1 ,2 . . . . . n be the n gyrostets of the system,each of mass MC whose

the centre Gi is defined in an inertial frame OXYZby the vector r~=(Ei,ni,~i).

We remind that a gyrostat is a mechanical system which is composed of one rigid body, called the carrier, and a numberofotherbodies,the

rotors, in connection

to the carrier. The individual massofeachrotoris either variably or constantly distributed in i t s volume, but the entire

mass of the gyrostat has the rigid

body property that i t s inertia components are time-independent constants. This 75

76

A.G. MAVRAGANIS

means that the motion of any rotor r e l a t i v e to the c a r r i e r

does not a l t e r the

d i s t r i b u t i o n of the total mass of the gyrostat [ 7 ] , [ 8 ] . The position r i in the system OXYZ of a point P i ° f t h e g y r ° s t a t ( g i

) is given

by the sum

= rG i + GiPi

r±

(I)

which may also be w r i t t e n {r i } : Here the t r i a d

[xi,Yi,Zi]~

{r~}+Ri[xi,Yi,Zi]

~'xiYiZi.

T (*)

determines the position

~ ~

fixed system GixiYiZ i of t~e p r i c l p a l

(2) of

Pi in

the body-

axes and Ri = RiRiR i , i . e .

~sOS(O4cos~-sin~isin@icos@~ [email protected]~.cos~.cos@. sin~.sin@:1 in~icos~ +cos~is in~icos@ ~ -sin~isin~i+cos~icos~icos @i-cos~is ine cosei z ~in~isin@~ - cos~isin@i

R. =

where e i , 8 i , ~ i

(3)

are the Euler angles.

In order to derive the equations of motion we need both the total

k i n e t i c and

potential energy, i . e . T=TI+T2+...+Tn and V=Vl+V2+...+V n where T.1 andV.1 are the respective contribution of (gi) into T and V. From Koenigs' theorem we readily obtain for Ti

-2 "2 ) + ½{uS~i{ui} + ½{ui}T{hi}+ T~z Ti : ~1M i ( [ "2 i + qi+
(4)

where J i is the diagonal form of the i n e r t i a tensor with elements the principal moments of i n e r t i a Ai , Bi and Ci , {u i } is the absolute angular velocity of (gi) and {h i } and T[ are respectively the t o t a l angular momentumandt~etotal k i n e t i c energy of the r o t o r ( s ) r e l a t i v e to the c a r r i e r . Supposing now that both {h i } and T~1 are known functions of the time { h i } = [ a i ( t ) ' 8 i ( t ), v i ( t ~T,. T i r = T [ ( t ) and expressing the angular velocity in terms of the Euler angles, i . e .

= [$isineisin~i+~icos~ i , "

sineioos%-els

%,

ico el

v

(5)

we find that

.? Ti:½M i (~i+qi+&i) "2 "2 ~2 + ~ [ ( A i s i n 2 , i + T ( A i c o s 2 * i + B i s i n 2 * i) + Ci

+ B i c o s 2 , i ) s i n 2 e i + Cioos2ei] + + ( A i - B i ) $ i S i s i n e i s i n * i c o s * i + Ci$i$icos8 i +

+-~-[(aiszn~ i 8icos~i)sinei+Vicosei] +T(aicos~i-Sisin~i)+ViT + T±

(6)

For V.z we consider two elementary masses dm i and dmj belonging respectively (*) The symbol { } denotes the matrix fo#m of a vecto# i.e. {a}:[a~,a2,a~] T

INTEGRAL INVARIANTS OF n GYROSTATS

to the gyrostats (gi) and ( g j ) a n d e v a l u a t e

the potential

77

function which causes

the mutual force acted on one another. We shall have dVij

= -~(Pji)dmidmj

(7)

where Pji Iri-rjl is the distance between the two masses. If we now integrate the above expression we take, after summing up for all j ( ~ i ) :

Vi

=j~iVij =j~iIMiI @(Pji)dmidmj ~Mj

(8)

where ~(Pji ) is a primitive of ~(Pji).We note that becauseof the transformation i J and on (2) each of the particular potentials Vii dependson both vectors r~,rG 1 n the triads (~i,@i,~i) and (~j,@j,~j). Hence Vi-Vi(r&,r ~2 .....r~,~l,81,~ 1 .....0n,

en,¢n). As both T and V have been found, i t is easy to derive the Lagrange's equations of motion f o r a l l gyrostats. Forweuse as generalized coordinates the variables Ei, qi,(i,~i, ei,~iandp ut q l i = $ i , q 2 i = q i , q3i=~i , q4i =~i' q5i=@i ' q6i=~i ; t h e r e f ° r e the desirable equations are d ~L dt'~qji

~L

i = 1,2 . . . . . n,

----0

j = 1,2 . . . . . 6

(9)

~qji

where L(=T-V) is the Lagrangian. ANALYSIS a. I n t e g r a l - i n v a r i a n t s of f i r s t order A p-tuple i n t e g r a l over a region in the s t a t e space of a system is c a l l e d a p-order i n t e g r a l - i n v a r i a n t i f i t has the same value f o r all times t [6]. As we

have said in the introduction the motion of the n gyrostats

is

controlled by

nine i n t e g r a l s , and whether @{hi}/~t= {0} and ~T~/~t =Obyonemorewhich is the

dacobi's i n t e g r a l .

The former integrals are:

n

i ~iMiqj i = Kj, n

j=1,2,3

(I0)

n

[ Miqji + [ M i ~ j i t : ~ j ,

i=l

j=1,2,3

(11)

i=l

n

[[Mi(q2iq3i-q3iq2i)+(Ai~li*ai)slnqsisinq6i+(Bi~2i+Bi)e°sq6~ i=l n

•

•

.

: Ul

+

[ [ Mi (q 3i q l i - ql i q3i )+(A i ~ l i +ai )sznq 5i cosq 6 i - (B i ~ 21 ~i ) s i n q 6 i ] = U2 i=l

(12)

n

[ [ M i ( q l i q 2 i - q 2 i q l i ) + ( A i = l i + a i ) c ° s q 5 i + Ci=3i * V~ : "3 i:l where K'S, A'S, ~'S are constants and ~ i j , j : 1 , 2 , 3 are the components of the

78

A.G. MAVRAGANIS

angular v e l o c i t y

{~i } expressed in terms of the Euleriananglesand t h e i r time-

d e r i v a t i v e s . The l a t e r , provided i t e x i s t s , is given by the expression n

~___LL)_k =h (const.)

i=l

~qli

(13)

~q2i

For convenience we shall denote the nine integrals (10)-(12) by fz(qji,~ji,t)=cc,

& = 1,2 . . . . . 9

(14)

where cL are constants and the l a s t one by

H(qji, qji )= h.

(15)

The above integrals furnish an equal number of

integral - i n v a r i a n t s , whose

general expression is [6] ~f~

afL

+

afc

af&

"

afz

"

(16)

where f l o = H ( q j i , q j i ). We use the l e t t e r ~ rather than d to emphasize that the increments are taken from a point of one o r b i t to the contemporaneous point of an adjacent o r b i t .

By D we symbolize the domain of the s t a t e - s p a c e over which

the integration is taken. I f t h i s domainisopentheintegral IC is called absolute and i f i t is closed the integral is called r e l a t i v e . From (10),(12) and (16) we find

I~ =

13+j

[ MiS~ji, £:j=1,2,3

07)

D i:l

: I {!lMi~qjl D ±

+ [n MitSqj i } ' i=1

j : 1,2,3

(18)

and three other expressions which correspond to the invariants the last three integrals.

In view of the derivation

integrals 13÷j , j = 1 , 2 ; 3 , the f i r s t

s i x invariants

of

the

obtained

second

11, 12 . . . . .

from

triad

of

16 are not all

independent. So, only three oftherm, e i t h e r (17) or (18) i t is worth to be used in the study of the system. Taking now into account the general

form

of

the

solutions of Eqs. (9) qji :

qji(q~i~ ' q-oj i ' . t ; t o ) ,

q j i = q j i ( q ~ i ' q-o. j i ' t ; to)

where q~i'] q~i] and t o are respectively the i n i t i a l

(19)

conditions and the i n i t i a l

time [5], we obtain, a f t e r performing the integrations over the original domain Do, the exact value of all

i n t e g r a l - i n v a r i a n t s I~,~=1,2 . . . . . I0. Notice that i f

we use several combinations of the integrals (14) we can derive a great number of other i n t e g r a l - i n v a r i a n t s of the same order with the invariants are only a part of the i n f i n i t e

previous ones.

These

first-orderintegral-invariantswhich

actually e x i s t in any dynamical system.We emphasize here that for the purpose of

INTEGRAL INVARIANTS OF n GYROSTATS

79

checking the accuracy of the numerical integration, their form must be suitably modified in order to be easy applicable whichever being the

numerical method

used. We close the presentation of this kind ofintegral-invariantswith a relative integral-invariant which is possessed by all

Hamiltonian systems. Fox we must

reformulate the problem in the system

dqji_ aH dt @Pji '

dpji = _ ~H dt ~qji

where Pji=@T/@~ji, j=1,2 . . . . . 6, i=1,2 . . . . ,n are the

(20)

conjugate momenta of the

coordinates q j i and H =

n

6

~

Z Pjiqji - L

i=z j=z is the Hamiltonian. Then, apart from the previous integral -invariants,

which

under their new form are also possessed by the system, there is a further one, whose the form is

111=

i i j~l pjiOqji =

oi=1 j~l pjiOqji

where C and Co are two arbitrary closed curves inthespace(qji,Pjj,, only once by the solutions of the system (20). This integral,

intersected

referred to as

Poincare's relative integral-invariant, shows that the "circulation" of the

n

gyrostats remains constant in time. In other words this means that during the motion the gyrostats do not change their translational and rotational behaviour. b. Integral-invariants of higher order So far we have spoken only of integral-invariants of order I, but we can also have integral-invariants of higher order 2,3 .....n. For Classical Dynamics the most important case are the extreme cases where the domain of integration has dimensions 1 or n. On the other hand exceptforthecorollarythatto any relative integral-invariant of order p corresponds an absolute

integral-invariant

of

order p+l, [6], it is very difficult to find direcNy otherinvariantsof higher order. Of course we could find a last multiplier M

and

then

to seek further

invariants by using the function M(xl,x 2 .....x6n,X6n+1) where x i, i=1,2 ....~n, stand respectively for the variables (qji,qji) or (qji,Pji) and X6n+z=t. This procedure is quite difficult, since the determination of Mrequiresto solve the equation

l d~M+6~+z @Xi - 0 M dt

i= I

Bxi

(22)

80

A.G. MAVRAGANIS

where X. are the second members of the equations of motion i . e . i

X~ = Xi(Xz'X2 . . . . . X6n;X6n+l)' X6n+l = X6n+l (=1) Thus we r e s t r i c t

(23)

our i n t e r e s t to the remaining extreme case of ordern. Like

all dynamical systems, the system in hand possessesan-order i n t e g r a l - i n v a r i a n t whose the general form is

where M=M(xz,x 2 . . . . . X6n, t) is a l a s t m u l t i p l i e r afore.

and the variablesx i a r e as in

I t is proved that i f the system is Hamiltonian the

last

multiplier

is

constant with value equal to unit i . e . M=I [6].Thereforethe i n t e g r a l - i n v a r i a n t I becomes

I=IIf x1 x2 x6n=ffI Xlo X2o0X no where x.

i0

denotes the i n i t i a l

Liouville's

value

of xi.Theaboveresultexpresses the known

theorem [9] that says that the area of any f i n i t e domain inthephase

space remains constant in time as the system moves

in

accordance

with

the

canonical equations. As a consequence, the degrees of freedom of the n gyrostats do not change during the motion, that in other words means that no changes occur in the dimensions of the phase-space.

REFERENCES [I]. Cid, R. and Vigueras, A.: Celest. Mech. 36, 155, (1985) ~. Duboshin, G.N.: Astron. J. U.R.S.S. 35, 2, (1958) [4. Duboshin, G.N.: Celest. Mech. 3/4, 423, (1971) ~. Duboshin, G.N.: Celest. Mech. 6, 27, (1972) ~. Pars, L.A.: "A Treatise of Analytical

Dynamics", Heinemann, London,(1965)

~. Whittaker, E.T.: "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies" Cambridge University Press. ~. Rumiantsev, V.V.: P.M.M.U.R.S.S.

25(1), 9, (1961)

~. Wittenburg, l . J . :

"Dynamics of Systems of Rigid Bodies", B.G.Teubner Stuttgard, (1977) ~. Synge, J.L. and G r i f f i t h , B.A.: "Principles of Mechanics", 3rd e d i t i o n , McGraw-Hill,Kogakusha, LTD, (1970).