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been used to reduce the order of the system by means of the integral of energy. Canonical equations of the reduced system so obtained are known as ...
I ntnat. J. Mh. Vol.

Math. S ci.

245

(1978) 245-253

GENERALIZED WHITTAKER’S EQUATIONS FOR HOLONOMIC MECHANICAL SYSTEMS

MUNAWAR HUSSAIN Department of Mathematics Government College Lahore, Pakistan

(Received June 23, 1977)

ABSTRACT.

In this paper the classical theorem "a conservative holonomic

dynamic system is invariantly connected with a certain differential form" is generalized to group variables.

This general theorem is then used to

reduce the order of a Hamiltonian system by the use of the integral of energy. Equations of motion of the reduced system so obtained are derived which are

the so-called generalized Whittaker’s equations.

Finally an example is given

as an application of the theory.

KEY WORDS AND PHRASES.

A:M.S(HOS) SUBJECT i.

Generalized Whlttaker’s equations, Holonomic Systems.

CLASSIFICATION

(19,70).

70F20

INTRODUCTION

It is known [4] that canonical equations for a conservative holonomic system whose Hamiltonian is H are obtained by forming the first Pfaff’s system of differential equations of the differential form,

Pidqi

H dr,

(i

1,2,---,n),

MLrNAWAR HUSSAIN

246 where

coordinates is implied.

are the generalized momenta corresponding to the generalized

Pn

PI’ P2’

ql’ q2’ ---’

qn

of the system and summation over a repeated suffix

This result leads to the theorem

connected with the differential form

"A dynamical

Hdt".

Pidqi

system is invarlantly

Further this theorem has

been used to reduce the order of the system by means of the integral of energy. Canonical equations of the reduced system so obtained are known as Whittaker’s

equations.

In what follows in this section we state a few basic results from

_

the theory of group variables in order to generalize the above mentioned results. Consider a conservative holonomlc system having n degrees of freedom and

whose position is specified by group variables x I,

x2,---,

be the parameters of real displacement and X I, X 2,---,

Xn.

Xn be

Let n I,

2,---,nn

the corresponding

displacement operators expressed by the relations

J

(i,J

Xl" i 8xj J where

i

are functions of x I, x 2, --,

Xn,

()

1,2, ---,n)

then for an arbitrary function

f(x I, x 2,---, xn, t) the infinitesimal change df is expressed by df

(2)

The X’s satisfy the relations

where

(Xo, Xi) Putting f

o(’f) + nixi(f)]dt

[X

xj

O,

(Xi, Xj)

CiJk

(3)

(k-,2,---,n).

in (2), we get

dx. ,j -.n dt Since the operators X

i

J

(4)

{i

are independent therefore the matrix

=o=-

singular and consequently (4) yields

ni

Aij xj

(5)

GENERALIZED WHITTAKER’ S EQUATIONS

24 7

Let L be the Lagranglan of the system then the canonical equations of

the

i, 2

system as obtained in

ri

H

Cjlk rj Yk Xi (H),

d--

---i’

(6)

1,2,---,n),

(i,J,k

where

are

L

n--

Yl and

H(Xl,

,Xn; yl,---,yn )

x 2,

nlyi

(7)

L

is the Hamiltonlan of the system and is equal to the total energy of the system.

2.

DERIVATION OF

CANONCAL EQUATIONS

FROM A

CERTAIN DIFERE..LkL, FORM.

In order to establish the invarlant relation between the system (6) and a certain differential form we prove the following theorem:

THEOREM.

The system of equations (6) is equivalent to the first Pfaff’s

system of differential equations of the dlfferentlal form

PROOF.

(nly2

H)dt.

We put

8d (niYi

H)dt

or using (5), we obtain 6

d

yiAij

d

xj

(8)

H dt

therefore

O where d and

x2, ---, Xn,

-

AiJ

x

(9)

H 6t

denote two independent variations in each of the variables x I,

YI’ ---’ Yn’

Od_dO= Yi[Aijdxj + t[dH

Yl

H dt]

t.

The bilinear corearlant of (8) is given by

Hyldt]+xk[yl AiJXk dxj X_k

t_dy

i

dxj] Aik-Yi Aik. xj (10)

MUNAWAR HUSSAIN

248 where we have used the relations

dx i dt

d x

(i- 1,2,---,n)

i

dt.

Equating to zero the coefficients of

I ’---’ n

---’

we get the first Pfaff’s system of equation in the form

H

Yl dt-

Aijdxj Yl

A

(i- 1,2,---,n)

0,

dYl Aik Yi BAlk dxj

H dt

dxj

H

dt,

dH

(Ii)

(i

0

(12)

(13)

i,2,--- ,n)

By vrtue of (5), the equations (11) asse the fo

H With the help of

dy i

d-q--

.

(i

(14)

1,2,---,n).

(4), the equations (12) become

J Ak Ym mi k

k

Ynm Aj x mi

8xj

k

k i

H

xk

which, by means of the relations (I) and (3), finally takes the form

dY__i dt

rj Yk

(15)

Cik XI(H)"

The relation (13) is a consequence of (14) and (15) and skew symmetric property

of

Cjik with

respect to the first two indices.

Since the equations

(14) and

(15) are identical with (6) the theorem is thus proved. 3.

GENERALIZED WHITTAKER’ S EQUATIONS Assume that H does not involve the time explicitly and H

/

h

is the integral of energy of the system.

the variable

Yl

(16)

O,

Let the equation (16) be solved for

so that it is algebraically equivalent to

K(Xl’---’ Xn’ Y2’---’ Yn’

t, h)

+ Yl

0.

(17)

GENERALIZED WHITTAKER’ S EQUATIONS

249

The differential form associated with the system is

(nl Yl + n2 Y2

+

---, xn,

where the variables x 1,

+

nn Yn + h)dt, h

Yl’ ---’ Yn’

are connected by (17);

the differential form can therefore be written as

where we can

+ rn Yn + h)dt (2 Y2 + n3 Y3 + regard (Xl,---, Xn, Y2’ ---’ Yn’ h, t)

(18)

n I K dt as the 2n+l variables

If we express (18) in the form

in the phase space.

Idt [ Y2

+

+

n Yn + lh_K] --i

(19)

and put

nldt then we take

T

dz

as the new time variable and

I___

i’

1

i

2

n2

i’---’ n

nn

i

as the parameters of real displacement, the corresponding displacement operators

and new momenta are respectively Xo, XI, ---,

Xn and h, YI’ --’ Yn"

Using the

result of section (2), the differential equation corresponding to the form (19) are

p 8K %yp’ dt d’r

8K

dyp dT

dh

)h

rl] Yk Cj pk -Xp(K)

(p

2 3---,n)

(20)

Oo

The last pair of equations can be separated from the rest of the system since

the first (2n-2) equations do not involve t and h is a constant.

The equations

(20) can be further simplified to take the form

K[Clpl +

nr Crpl] + Yr Clpr+ nrYq Crpq Xp(K) (p,q,r

2,3,---,n).

(21)

MUNAWAR HUSSAIN

250

The original differential equations can therefore be replaced by the

reduced system (21) which has only n-I degrees of freedom.

The equations (21)

are the desired Whittaker’s equations.

4.

AN EXAMPLE Consider a rigid body which is moving about one of its fixed points 0

We introduce a fixed frame of reference Oxyz

under the action of gravity.

such that Oz is vertically upwards and a moving frame Ox’y’z’ which coincides with the principal axes of inertia of the body at O.

Eulerian angles 8, and

,

Let us choose the

(8 is the angle of nutation,

the angle of precession

the angle of proper rotation) as the group variables which specify the

Obviously the dynamical system under con-

position of the body at time t.

slderatlon is a conservative one and it has three degrees of freedom.

Choosing

the parameters of real displacement as the components of angular velocity

along the moving axes, we have the relations

.n2

os

+

--Sln

+

$

co,

sin

Sin 8 Cos

_ -+

Consequently the displacement operators

X1

X2

x3

sin

+ Cosec e

Cos

Sin

+

Cosec

,

sin

e

Cos

X2

X are given by 3

Cot

e

Cot

sin

e

Cos

(22)

which satisfy the commutation relations

XlX2-X2X1 X3 X3) X2X3 X3X2 X1 X1) X3X1 XlX3 X2

(X1, X2) (X2, (X 3,

(23)

GENERALIZED WHITTAKER’ S EQUATIONS

C’s are therefore expressed by the

The non-vanlshlng

C213 C321 C132

C123 C231

C312

251

relations

i,

1,

(24)

i.

Let T and U denote the kinetic and potential energies of the system respectively, then

(A n + Bn 2 +

T

Cn3), (25)

+

Mg(x Sln B Sin

U

te

where A, B, C are

y Sin B Cos

+ z Cos B

principal moments of inertia at O; x, y, z are the

coordinates of the centre of gravity of the body wth respect to the moving

-

Using (25), we have the Lagranglan L and

axls and M Is the mass of the body. momenta

Yl’ Y2’ Y3 U

T

L

A

Yl

expressed by the relations:

(An + Bn + Cn ) + Mg

1’ Y2

B

2’ Y3

(

Sin 0 Sin

+y

Sin O Cos

(26)

n 3.

C

+z Cos O),

In view of (26) the Hamiltonlan H is given by H

=

Using

+__+ "A

Mg(x Sin e sin

+ y

Sin

e

Cos

+

z Cos e)

(27)

B

(6), (22), (24), (26), and (27), canonlcal equations of the system are

n1

Y1, n 2 A

dYl

B-C BC

dt

dY2 dt

dY3

C-A CA

Y2, n 3 B

73 C

y2Y3 + Hg (" Cos e

y3yI + Mg (-

A-__B yly2 + Mg

Sin

Cos O

"

Sin

-

e

Cos ),

(28)

+ Sin e

e(x Cos

Sin

)

y Sin ).

252

MUNAWAR HUSSAIN

Now the relation (16) gives

--+

----+--- 2Ms(

+ y

Sin 6 Sin

Sin 6 Cos

+

z Cos 6)

+ 2h

0,

and consequently

A[2HS( Sin

Yl

,

0 Sin

+

Sin 0 Cos

+

Cos 0)

-__2 -__Y

2hi,

Comparing this relation wth (17), we get

A[2ME("

K

Sin

e

+

Sin

e

y Sin

Cos $

+

..___-

z Cos e)

B

2h]

;.

C

Therefore by the application of (21) the canonical equations of the system reduce to 3K

2

Y2

----dY2 d’r

Y3

dY3

r3

3K

’3’ X2(K),

n3 Yl Y2 + n2 Yl

X3(K).

Now

dY2 dY2 d

dt

dY3 dY3 de

dt

K h

K 9h

dY2

A K dt

dY3

A K dt

=. Mg(-x Cos x3 ) " t s cos X2(K)

0

+

z Sin

e sla ),

s

s e.

)

(29)

GENERALIZED WHITTAKER’ S EUATIONS

25 3

Therefore equation (29) assume the form

r _._.dY2 dt

dY3 dt

A

BY2’

rl

A

-y3

KA_._y3 + Ms(-’Cos e +

e

Sin

,),

-’"

Sin

).

Sin

CA

K(B-A)

Y2 + HS;

Sin

e(" Cos

These are the Whittaker’s equations for the system under consideration. REFERENCES

1.

Cetaev, N. G. On the Equations of Poincare, Prlkl. Mg. t...Me.h.(1941) 253-262.

2.

Hussain, M. Hamllton-Jacobi Theorem in Group Variables, Journal of Applied Mathematics and Physics (ZAMP), Vol- 27, (1976) 285-287.

3.

Poincare, H. On a New Form of the Equations of Mechanics, C. R. Acad. Sci. 132 (1901) 369-371.

4.

Whittaker, E. T. Analytic.al Dynamics of Cambridge University Press, 1961.

Particle.s

and Rigid Bodies,