Mathematical Methods for Physics and Engineering : A Comprehensive Guide

9.2 SYMMETRY AND NORMAL MODES

M M

M M

k k

k

k k

k

x 1

y 1

x 2

y 2

x 3

y 3

x 4

y 4

Figure 9.4 The arrangement of four equal masses and six equal springs

discussed in the text. The coordinate systemsxn,ynforn=1, 2 , 3 ,4 measure

the displacements of the masses from their equilibrium positions.

|Rm−Rn||qm−qn|,weobtainbmn(=bnm):

bmn=^12 k

[

|(Rm−Rn)+(qm−qn)|−|Rm−Rn|

] 2

=^12 k

{[

|Rm−Rn|^2 +2(qm−qn)·(RM−Rn)+|qm−qn)|^2

] 1 / 2

−|Rm−Rn|

} 2

=^12 k|Rm−Rn|^2

{[

1+

2(qm−qn)·(RM−Rn)

|Rm−Rn|^2

+···

] 1 / 2

− 1

} 2

≈^12 k

{

(qm−qn)·(RM−Rn)

|Rm−Rn|

} 2

.

This final expression is readily interpretable as the potential energy stored in the

spring when it is extended by an amount equal to the component, along the

equilibrium direction of the spring, of the relative displacement of its two ends.

Applying this result to each spring in turn gives the following expressions for

the elements of the potential matrix.

mn 2 bmn/k

12 (x 1 −x 2 )^2

13 (y 1 −y 3 )^2

1412 (−x 1 +x 4 +y 1 −y 4 )^2

2312 (x 2 −x 3 +y 2 −y 3 )^2

24 (y 2 −y 4 )^2

34 (x 3 −x 4 )^2.