131
Internat. J. Math. & Math. Sci.
(1986) 131-136
Vol. 9 No.
A GROUP THEORETIC APPROACH TO
GENERALIZED HARMONIC VIBRATIONS
IN A ONE DIMENSIONAL LATTICE
J. N. BOYD and P. N. RAYCHOWDHURY
Department of Mathematical Sciences
Virginia Commonwealth University
Richmond, Virginia 23284 U.S.A.
(Received July I0, 1984)
ABSTRACT.
Beginning with a group theoretical simplification of the equations of
motion for harmonically coupled point masses moving on a fixed circle, we obtain
By taking the number of vibrating
the natural frequencies of motion for the array.
point masses to be very large, we obtain the natural frequencies of vibration for any
arbitrary, but symmetric, harmonic coupling of the masses in a one dimensional lattice.
The result is a cosine series for the square of the frequency,
f2j
i
--F
s
a()cosE where 0
2j
N
B
=0
the attractive
......
{1,2 3,
2, j
N} and a() depends upon
force constant between the j-th and (j+g)-th masses.
Lastly, we show
that these frequencies will be propagated by wave forms in the lattice.
KEY WORDS AND PHRASES.
Harmonic coupling, frequencies
AMS SUBJECT CLASSIFICATION CODE.
i.
of motion, wave forms,
lattice.
20C35.
iNTRODUCTION.
In this paper, we suppose that N identical, point masses are symmetrically and
uniformly arranged around a fixed circle so that the masses are coupled with massless,
ideal springs.
All motions are confined to the fixed circle.
We write potential
energy matrices and show how to diagonalize these matrices when the symmetry is that
of the rotational group of order N.
Having begun with quite simple examples, we
next take N to be very large to obtain the natural frequencies for a one dimensional
crystal lattice in which both nearest neighboring point masses and next nearest
neighbors are coupled by harmonic forces.
We then extend this result to include all
symmetric couplings and show that the lattice will support wave disturbances of pre-
cisely the natural frequencies of the lattice.
It should be noted that our results
follow from the consideration of the interactions of all pairs of particles around
the circular lattice rather than from a Fourier expansion of the potential energy of
interaction.
This work represents a further use of group theorectic methods which have
been reported elsewhere [1,2,3,4,5].
2. POTENTIAL ENERGY MATRICES.
Let us consider the symmetric N
manner: N
_> 3; v..j3
2 for j
N matrix V(N,s)
{1,2,3,...,N}; Vjk
(Vjk)
-I for k
defined in the following
j
-+ s(mod N) where
152
s e
and
J.N. BOYD and P. N. RAYCHOWDHIRY
{1,2,3,..[lN--!l]}
[l--!l]
and
+/- the greatest integer less than
or equal to
--;
0 -i
0
0 for all other entries of V(N,s).
Vjk
For example,
2 -i
V(3,1)
V(5,1)--
2-i
0 -I
0
0-1 2-1
0 -I 2
0 0 -I
and V(6,2)--
0
10
The matrix V(N,s)
2
-12
0 -i
-1 0 2 0-1
0 -i 0 2 0
-i 0 -I 0 2
0 -I 0 -i 0
-i
arises in writing the elastic potential energies of a symmetric
circular array of N point masses interconnected with ideal springs [i].
If each point
mass is coupled to both nearest neighbors but to no other masses, then s
i.
If each
point mass is coupled to its two next nearest neighbors but to no others, then s
Consider the case for N
frictionless circle.
4, s
2.
Four identical masses move on a fixed
i.
They are connected by springs of force constant k as shown.
x
I
x
x
4
2
x
FIGURE I.
Let the coordinates,
Xl,X2,X3,X 4,
from their equilibrium positions.
elastic potential energy is
i
2
P.E.
X
k
x2
x3
x
[(Xl-X 2)
+
(x2-x 3)
2
+
denote very small displacements of the masses
All motion is confined to the circle, and the
(x3-x4)
2
+
(x4-x I)
2
1
k
V(4,1)X
where
X, and V(4,1) is as defined.
is the transpose of
4
We can represent the system with a graph on four vertices.
The vertices repre-
sent the point masses, and two vertices are connected by an edge if and only if the
two corresponding masses are coupled by springs.
1
4
2
FIGURE 2.
133
GROUP THEORETIC APPROACH TO GENERALIZED HARMONIC VIBRATIONS
Thai is, each mass is coupled only to its two next
2.
5, s
Suppose that N
We emphasize that the springs lie along the
nearest neighbors as indicated below.
circle and all motion is on the circle itself.
i
2
3
FIGURE 3.
If the springs have force constant k, we have
1
P.E.
k
- -- -_ -
[(Xl-X 3)
2
+
(x2-x 4)
2
+
(x3-x5)
2
+
(x4-x I)
2
+
(x5-x2) 2]
1
k
V(5,2)X.
GROUP REPRESENTATIONS AND UNITARY TRANSFORMATIONS.
3.
2
As rotations by
leave the system of N point masses and springs unchanged in so
far as kinetic and potential energies are concerned, we take the rotation group
{R,R
C(N)
2,...RN}
where R denotes a rotation of the circle by
As the group is abelian, all irreducible, nonequivalent matrix
group of the system.
representations of C(N) are one dimensional
Thus the N cyclic groups under complex
exp(-),
multiplication which have generators
representations
to be the symmetry
N},
{1,2,3
give all irreducible
[2].
From these representations we can construct the unitary transformation matrix U
which will simultaneously diagonalize
"PT T
"P-exp
U
N
exp
V(N,s) for all s:
4i
exp
6i
exp
8i
exp
12i
12i
18i
@xp
exp
4(N-1)__t
N
exp
6(N-1)_t
N
Since much of what is to follow depends upon this fact, the statement should be
justified.
1
(exp
4 exp
Consider the transformed matrix
exp
sin
2 rsj
N
Recalling that U
/4 exp
N
N
The j-th row of UV(N,s) is
(4 exp
2 sj
N
/4 exp
2
2Njt
sin
N
sin
2
sj
N
is simply the complex conjugate of the transpose of U, we
can write the j, k-th entry of
(_4N) sin2
sin
-1
1)V(N,s)
exp
exp
-I
UV(N,s)U
N
=i
exp
[UV(N,s)]U -I-"
2(j-k)iN
The claim has been justified, and UV(N,s)U
This entry is
"4 sin2
zsJ
for
for j
k,
# k.
-I is diagonal for each s.
J.N. BOYD and P. N. RAYCHOWDHURY
134
AN APPLICATION.
4.
Let us consider a circular arrangement of N identical, uniformly spaced particles
For very large N, we have, in effect, applied the Born condition to trans-
of mass m.
form a linear, one dimensional crystal into a circular array [2].
Suppose that near-
est neighbor particles are coupled with ideal, massless springs of force constant
k(1) and that next nearest neighbors are coupled with ideal, massless springs of
force constant k(2) as indicated below.
\
/
FIGURE 4.
This configuration would serve as a model for a physical lattice in which harmonic
forces between next nearest neighbors were much larger than anharmonic forces between
In the event that the reader does not wish to visualize such a
nearest neighbors.
mechanical arrangement, he could easily interpret the problem in terms of identical
LC circuits with capacitors coupled in the appropriate manner.
The Lagrangian for the system is
[ k(1) C(N,1)X
L
k(2)
(N,2)X]
N identity matrix,
where I is the N
and X
+
(XlX 2
).
We can now transform the Lagrangian to diagonalize
N
-i (UIU-I)u[- [--k(1)
=(UV(N,2)U-I)ux]
1
V(N,I) and V(N,2): e
+
1
k(2)
U
-i
0
sin
sin
0
0
0
0
2
The column matrix
I
m
0
0
0
0
sxn
1/2k(2) *
3
0
0
N
iU -i (UV(N,I)U
N.
0
0
sin
0
o
41"
0
0
0
n
sin
i!
N
o
o
UX gives the new symmetry coordinates in which all the
potential energies are uncoupled
transpose of
sin
I)UX
The row matrix
is the complex conjugate of the
The equations of motion are given by
0
0
GROUP THEORETIC APPROACH TO GENERALIZED HARMONIC VIBRATIONS
where
have
j
mj
is the j-th symmetry coordinate and
2
4(k(1)sin J_i
N
+
+ k(2)sin
2
is its complex conjugate.
j
Thus we
N}.
{i 2 3
0 for j
)nj
135
The natural frequencies are then
/k(1)sin
fj =--I 4
j_ + k(2)sin2 2j
N
N
This expression reduces to the well known result for the linear lattice for k(1) >
O,
0[2].
k(2)
AN EXTENSION.
5.
It should be clear that if the N point masses are coupled with force constants
k()
e{1,2,3
0 and potential energy matrices V(N,) for
s}, then the J-th fre-
quency would become
1
/i
7j=i [k() sin2
fj
]/m
In this event, we can write
k() (i
=1
j
k()sin2
=i
k()
a()
))
.[
s}
and 0 <
be called the wave number of the vibration.
absolute convergence for
k()2
a()cosB where a(0)
=0
e {i 2,3,
for
2
cos(
I
2j
N
B
If we let N
2.
The variable 8 will
and s
while assuming
we have a model in which we can take the harmonic
=l
vibrations for all possible symmetric couplings to approximate some frequency distri-
bution obtained from an arbitrary potential energy of interaction
Supposing that f
1
3
is computed or found experimentally to be
then we can obtain a() by noting that
4m
1
and a()
12
g()cosSd.
a()cos2Bd8
g() cosB d8
0
0
Since, the frequency function should be symmetric with
0
respect to
jN
our model will be consistent for g(8) symmetric with respect to
B
7,
6.
LOCALIZED DISTURBANCES AND WAVE PROPAGATION.
thus explaining the absence of sine terms in the Fourier series.
The frequencies obtained are those for the lattice moving as a whole.
That is,
the frequencies are those of the normal modes of oscillation which involve motions of
all constituent parts of the lattice at once.
We now return to the original coordinates in our circular array.
for j e
{1,2,3
.
,N}
equilibrium position.
k(s)
0 if s #
where
x.
3
take "the force constant k() > 0 with
O, the j-th particle is disturbed. The disthe site of the (j + )-th particle until some time
For the moment,
letus
Suppose that at t
If we postulate that a wave front will move out from the site of the j-th
particle at t
xj
is the displacement of the j-th point mass from its
turbance will not be registered at
later.
These are
.
0, then the time required for the front to reach the (J + )-th
particle will be proportional to
J +
J
136
J.N. BOYD and P. N. RAYCHOWDHURY
Taking the proportionality factor to be b, we
express this Dhysical observation
by the introduction of a phase factor e 2bji/N
to multiply
the time dependent part of
the displacement of the j-th particle
from its equilibrium position. That is, we
make
2bji/N
the change of variable
e
Then
x.(t)3
2bEi/N
e
yj(t).
xj+(t)/xj(t)
yj+E(t)/yj(t)
implies a difference in phase equal to
Ibl as required.
1
1
From the Lagrangian in the original coordinates, L
m XIX +
k
we obtain the equation of motion m.
k(%)[-2x +
+
xj+
j
If we now suppose that a periodic wave moves through the array, we
write for each
i/N
2vbj
2bj
2fit
e
j,
e
e
where Y is a constant amplitude. Then the
()V(N,)X,
xj_].
i/Ny
yj(t)
x.3
equation of motion becomes
2e2J i/N
f2
k()
2
mj
-m(2f)2ye 2bji/Ne2fit= k(g)ye2fit(e 2b(j+)i/N
+ e 2b (j-)i/N ),or m(2f) 2
.b.
sin2(--)
k()
(I
(e2b li/N 2 + e -2bi/N) Therefore
i /()(l-cos
2b
2b
or f
where
cos(N)
j
-k()
Thus a longitudinal traveling wave having freuqency of any of the normal modes
(k() > 0, k(s)
0 for
s) will be supported by the lattice.
Similarly, from the general Lagrangian in the coordinates
.,
e
i
m
I
+
1
3
s
E()V(N,)X,
s
we obtain
=i
m.3
+
k()[-2xj
--
.
xj+ + xj_E]
implying the existence of a traveling wave form constructed from the superposition of
all waves for some fixed b and with the summation taken over
ye2ji/N e 2fit
2
1
s
s
i
We have x.
[ a )cosS
where f
[ k()sin2 b
3
N
=2m =0
m =i
with a() and 8 as previously defined, provided that b is an integer.
We conclude with a change to a notation more familiar in solid state physics.
Let u be the equilibrium spacing between successive particles in our array, let
0
2
ye<Ui e 2fit Ye (<ui+2=fit) and, for
ju
u, and let <
Then x.
x(u)
0
Nu
J
0
fixed b, x(u) satisfies the wave equation
2x 2 x
(42f2
2-)---=----"
u
the wave is
/
__if
_2
<
<
7N
Then the velocity of
t
s
[
=0
a()cos
s
REFERENCES
i.
BOYD, J.N. and RAYCHOWDHURY, P.N., Representation Theory of Finite Abelian Groups
Applied to a Linear Diatomic Crystal, International Journal of Mathematics and
Mathematical Sciences 3 (1980) 559-74.
2.
BOYD, J.N. and RAYCHOWDHURY, P.N., An Application of Projection Operators to a
One Dimensional Crystal, Bulletin of the Institute of Mathematics, Academia
Sinica 7
(1979) 133-44.
3.
BOYD, J.N. and RAYCHOWDHURY, P.N., A One Dimensional Crystal with Nearest Neighbors
Coupled Through Their Velocities, ASME Journal of
Systems Measurement
and Control 103 (1981) 293-6.
4.
BOYD, J.N. and RAYCHOWDHURY, P.N., Group Representations in Lagrangian Mechanics,
Physica I14A(1982) 604-8.
5.
BOYD, J.N. and RAYCHOWDHURY, P.N., Two Dimensional Lattice Vibrations from Direct
Product Representations of Symmetry Groups, International Journal of Mathematics and Mathematical Sciences,
6__(1983) 783-94.