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,