I nternat. J. Math. & Math. Sci.
Vol. 3 No. 2 (1980) 369-382
369
PROJECTION OPERATOR TECHNIQUES IN
LAGRANGIAN MECHANICS: SYMMETRICALLY COUPLED OSCILLATORS
J.N. BOYD and P.N. RAYCHOWDHURY
Department of Mathematical Sciences
Virginia Connonwealth University
23284
Richmond, Virginia
U.S.A.
(Received July 6, 1979)
ABSTRACT.
We apply projection operator techniques to the computation of the
natural frequencies of oscillation for three symmetrically coupled mechanical
systems.
In each case, the rotation subgroup of the full symmetry group is used
to determine the projection operators with the result that the Lagrangian must be
expressed in terms of complex-valued coordinates.
In the coordinate system obtained
from the action of the projection operators upon the original coordinates, the
Lagrangian yields equations of motion which are separated to the maximum extent
made possible by symmetry considerations.
KEY WORDS AND PHRASES. Group representations, Harmonic
mechanics, Natur frequencies, Projection operators.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
osors,
Lagrangian
20C35, 70J10.
INTRODUCTION.
We consider a vibrating mechanical system of point masses and ideal rotators.
J. N. BOYD AND P. N. RAY CHOWDHURY
370
X
If we define the n-dimensional column vector,
we are able to write the Lagrangian for the system as
col(XlX2"’’xn)
X
mT
1
L
where
have
X
X
the transpose of
i,2,-..,n
and
1
kXVX
is the row vector
Xl,X2,.--,xn
n x n
Let
X-
Here we
(XlX2-..xn)
as velocities and displacements with
{
respect to an orthonormal set of vectors,
are symmetric
by the notation
Both
l’U2 ’’’’’un
T
and
V
matrices with all entries real.
be the finite point symmetry group of the mechanical system, and
G
let us denote the distinct, nonequivalent, unitary matrix representations of
G
by
F
()(G)
classes in
the matrix
G
e{l,2,---,k}
where
geG
Then for
r()(g)
is the number of conjugacy
k
let us denote the
i
j-th complex entry in
F..()(g)
by
#(g)
Further, let
and
be the linear operator induced by
{Ul’U2’’’"Un }
solution space which has basis
geG
acting on the
Then we are in a position
to define the projection operator
Pij
[G
There are exactly
Z rij () Cg) O(g)
()
geG
of these operators
[4], [S]
{u u -’-,u }
1 2
n
The action of these operators on the basis
will produce
a new basis of symmetry coordinates wth which it is possible to exploit the
geometric symmetries of the system to the maximum extent in the solution of
the equations of motion
a unitary matrix
U
[6]
That is, from the new coordinates, we construct
such that
UTU -i
and
UVU
-I
will assume block diagonal
forms which lead to significant simplifications in the equations of motion.
In general, the entries in
U
will be complex, leading to transformed
velocities and displacements with real and imaginary components.
371
PROJECTION OPERATOR TECHNIQUES IN OSCILLATORS
That is,
L
N
where
1
1
U-Iuvu-I UX
U-1UTU-1U k
m
1
*
-mUTU-I
kl UVU- IN*
col(qlq2.-.qn)
j
Note that
and
qjcC
and that they give the
Also observe that
qj*
denotes the complex conjugate of
[1], [2].
e.
velocity and displacement with respect to the symmetry coordinate
Bj
The equations of motion are then given by
0
d
by
}L
or
equivalently,
}L
The use of the Lagrangian in the case that the new coordinates are complex
is both convenient and intrinsically interesting.
2.
EXAILES.
We now give three problems to illustrate the technique.
EXAILE I.
have the value
below.
Six point masses of which three have the value
m
and
three
M are symmetrically arranged about a fixed circle as shown
They are interconnected with identical springs of force constant
k
We desire to find the natural frequencies of vibrations for this system if all
motion is of very small amplitude and is confined to the fixed circle.
SOLUTION.
We take the symmetry group to be
which is a subgroup of the full symmetry group
The irreducible representations over
C
C
3
{E, R(120 )
3v
are given by
R(240)}
J. N. BOYD AND P. N. RAY CHOWDHURY
372
r C1) (c3)
r(1)(g)
1
gC 3
F(2)(C3)
F(2)(E)
1
F(2)(R(120 o))
r(2) (R(240))
F
(3)(C3)
F
(3)(E)
i
r (3) {R(240))
..
.
1
2
F
i/
2
(3)(R(120 ))
-
1 +
2
i
2
I +
=
There are three projection operators which act upon the basis of unit
tangents.
pC1}
11
p{2)
11
pC3)
11
r{1} {g)Cg)
gcC 3
r {2) {g)0{g}
and
FC3) Cg)O{g)
where
gcC 3
gcC 3
PROJECTION OPERATOR TECHNIQUES IN OSCILLATORS
is the linear operator corresponding to the symmetry element
373
By computing
g
the effects of the projection operators upon the unit tangents,
uj
we form
a new basis for our solution space and we can construct the unitary matrix
with which we shall transform the Lagrangian.
"(1)Ul
Vll
Pll-(1)u2
P(2)Ulll
P(2)u211
u
Pll(3) 1
P
+ u
I
+ u
u2
+
Ul
1
+ (2
u2
+
u4
(_
1
"
(1
+
u2
+
S
u6 -a
i
u3+ (_i
(- 2
+
Ul
u2
3
We construct the matrix
+
i
"U
+
US
(- +
u4
+
u3
1
+ (_
2
i
u4
1
+ (2
i
u6
us
u6
after normalizing the six orthogonal
vectors above.
U=
and
1
0
0
1
1
0
0
1
1
0
(-+
0
1
0
1
0
0
1
0
1
0
(-/
0
0
(-+
(--
0
-0
.(
1
i
1
1
0
1
)
"_1
1
0
1
(-+
.)
0
1
(-
U
J. N. BOYD AND P. N. RAY CHOWDHURY
374
1
0
1
0
1
0
1
0
1
0
1
0
C-+
0
0
1
0
1
0
(---
0
1
0
0
1
0
(-2
The kinetic energy matrix is
and
UTU
-I
0
0
0
0
0
0
0
0
1
0
0
0
0
M/m
1
0
0
0
M/m
0
0
0
1
0
0
0
0
0
0
0
T
The potential energy matrix
is
2
-I
0
0
0
-I
-I
2
-I
0
0
0
0
-I
2
-i
0
0
0
0
-I
2
-I
0
0
0
0
-I
2
-I
-I
0
0
0
-I
2
and
PROJECTION OPERATOR TECHNIQUES IN OSCILLATORS
2
-2
0
0
0
-2
2
0
0
0
o
o
2
(-+
o
2
o
2
UVU -I
o
Thus
L
if
1
g- ---)
(-
0
0
0
o
o
o
col(qlq2q3q4qSq 6)
N-
and N*
1
1 UTU-I,
k UVU-IN.
m
!k2 (2nln*1
1
col
",
(miD I + M2
2nln 2n2n*1 + 2n2n + 2n3n + (-
375
*
*
(ln2n3n4nsD6)
+
m3
+
+
nsn4
we have
MD4D + mD5 + M6)
2
The equations of motion are given by
L
d. 8L.
-t--&-)
o,,3
je{1.2,3.4,5,6}
0
Thus we have
1
2k
W ril
f3
f4
and
4
2
2"
n6
2k
0
k
2k
2 lll
and
n2
fl
frequencies
and
+
and
f2
1
2--
2k
ier)n3 -4
(1
2k
rl2
/2k (
1 + 1
)
implying natural
3
2k
m
r3 +
k
implying natural frequencies
1/2
and
ml_ ) (.1m2 Mn1 )
M
/(1 ) Cm_2"1 N1 M1.)2 112 ;fl5 2km n5
2k
k
-(I ir) 5 ---n 6 implying that f5 f3
+
+
+
+
+
+
(1-i) B3
+
and
f6 f4
J. N. BOYD AND P. N. RAY CHOWDHURY
376
EXAMPLE 2.
Three identical rods of length
are pivoted about a fixed
common axis at the centroid of an equilateral triangle so that the rods lie
along the angle bisectors as shown.
The ends of the rods which fall on the
vertices of the triangle are connected with identical springs of force
k
constant
The moment of inertia with respect to the fixed axis is
We want to compute frequencies for very small amplitude
for each rod.
vibrations in the plane of the triangle.
-
SOLUTION.
P E
where
uj
1
X
and
k
E
(x 1
The potential energy is given by
x2
col(XlX2X3)
2
+
(x 2
xj
x
S)
2
+
(x 3
Xl)
2-[
cos
1
20o
6
-k ,VX
is the displacement of the vertex along
I
377
PROJECTION OPERATOR TECHNIQUES IN OSCILLATORS
V
The kinetic energy is
2
4
4
1
4
1
2
1
4
1
4
1
4
1
2
1
K.E.
I
T
0
0
+
I
0
0
1
0
0
1
operators are as in the preceding example.
and
e3
23
C
1
(
+
=
u2
1"
(Ul
+
u3)
/
(-Y +
{Ul
e2
1
__)"u2
1
1
1
2
+
2)2 + {=+
i/.)S
Thus
1
1
(-2
C=
1 +
"
(-
"-)
where
Their action upon the basis
{- 2
+
T
and the projection
3
produces the syetry coordinates
{Ul
el
+
2
The symmetry group is again taken to be
{Ul,U2,U 3}
X2
’+
1
2
i,)
J. N. BOYD AND P. N. RAYCHOWDHURY
378
Next we obtain
0
UTU
-I
T and UVU
-i
0
0
-0
0
0
0
Then the Lagrangian is given by
L
3
1 1
I
(-n2n
G2n }3I) -k
-(--)(Gln[
+
+
The equations of motion are
39,,2k
41
fl
0
f2 f3
rl 2
and
32k
+
3
n3n
implying the frequencies
4-
Lastly, we present a problem similar to that in the first example.
EXAMPLE 3.
Eight point masses of which four have the value
four have the value
M
m
are symmetrically arranged about a fixed circle.
The masses are interconnected with identical springs of force constant
Motion is again confined to the circle and we wish to find the natural
frequencies.
and
k
379
PROJECTION OPERATOR TECKNIQUES IN OSCILLATORS
SOLUTION.
C
4
If we take the symmetry group to be
(E,R(90), R(180), R(270)}
(1)(c 4)
F (1)(g)
1
F C2) (C 4)
‘(2) (E)
r
F
’(2) gC
r (3) (.)
4
(R(180 ))
r (3) (R(90 )
1
F (3) (R(180))
"(4)(C 4)
1
"(2)(R(270 ))
(2)(R(90 ))
r (3) (C 4)
there are four irreducible representations"
1
1
i
r (3) (R(270))
"(4)(E) 1 "(4)(R(90 ))
i
p(4)(R{270 o))
F (4)(R(180 ))
-’1
i
i
There are four projection operators which determine the
transformation matrix
U
They are
pC1)
y.
11
gC 4
p(2)
11
..
p(3)
11
p(4)
11
l, (1)
(g) qb (g)
i., (2)
(g)(g)
gEC 4
F(3) (g) ) (g)
and
gC 4
r (4) (g)qb(g)
gC 4
After the projection operators are applied to each unit tangent vector,
the resulting, distinct, nonzero vectors are normalized to give the rows in
the matrix
U
Thus
J. N. BOYD AND P. N. RAYCHOWDHURY
380
U
1
1
0
1
0
1
0
1
0
0
1
0
1
0
1
0
1
1
0
-I
0
1
0
-1
0
0
I
0
-I
0
1
0
-1
1
0
-i
0
-I
0
i
0
0
i
0
-i
0
-1
0
i
1
0
i
0
-i
0
-i
0
0
1
0
i
0
-i
0
-i
The Lagrang
is
ml
L
__T_
-k VX
where
X--col
1
(XlX2X3X4XsX6X7X8)
1
0
0
0
0
0
0
0
0
M/m
0
0
0
0
0
0
0
0
i
0
0
0
0
0
0
0
0
M/m
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
M/m
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
M/m
2
-i
0
0
0
0
0
-I
2
-I
0
0
0
0
0
0
-I
2
-i
0
0
0
0
0
0
-i
2
-I
0
0
0
0
0
0
-I
2
-i
0
0
0
0
0
0
-i
2
-i
0
0
0
0
0
0
-i
2
-1
-i
0
0
0
0
0
-i
2
T
and
V
Then
UTU -I
T
and
PROJECTION OPERATOR TECHNIQUES IN OSCILLATORS
381
2
-2
0
0
0
0
0
-2
2
0
0
0
0
0
o
0
2
0
0
0
0
0
0
2
0
0
0
0
0
0
2
(-l+i)
0
0
0
0
0
(-1-i)
2
0
0
0
0
0
0
0
2
(-l-i)
0
0
0
0
0
(-l+i)
2
The transformed Lagrangian is
-
-(mn In-.1
1
L
l
+
+
k
+
M2 m3 M4
(2rl rl I
(-l-i)n6n
+
+
2r2r.- 2rlr
+
(-l-i) n7n 8 +
(-l+i)ns
+
+
+
2r2r
ms
+
+
6 m7 } 8)
+
+
2r3r 3 + 2r4r + 2nsr
26 .6 + 27n7" + (-i+i)8’7
2n8 )
The equations of motion and natural frequencies are found in the standard
manner with the results that
fl
0
f2
1
2-’
/2k (
1 + 1
f5 f7
f6 f8
1
(i
)
+
)
+
)
1
f3
+
m2
M2
and
1 1/2
(i__2 + )
m
REFERENCES
I.
Boyd, J.N. and P.N. Raychowdhury, "An Application of Projection Operators
to a One Dimensional Crystal," Bulletin of the Institute of
Mathematics, Academia Sinica 7 (1979), 133-144.
382
J.N. BOYD AND P. N. RAYCHOWDHURY
2.
Duffey, G.H., Theoretical .Phy.sics, Houghton Mifflin Company, Boston, 1975.
5.
Gamba, A., "Representations and Classes in Groups of Finite Order,"
Journal of Mathematical Physics 9 (1968), 186-192.
4.
Hamermesh, M., G.roup..._Theory .and Its Application to Physical Problems,
Addison-Wesley Publishing-Company, Reading, Mass., 1962.
5.
Joshi, A.W., Elements of
New York, 1973.
6.
Nussbaum, A., "Group Theory and Normal Modes," American Journal of
P_sics 36 (1968), 529-539.
7.
Wilde, R.E., "A General Theory of Symmetry Coordinates," American
3ournal of_Physics 32 (1964), 45-52.
Group .Theorr
for Physicists, Halsted Press,