Mathematical Methods for Physicists: A concise introduction - Site Map

NORMAL MODES OF VIBRATIONS
Figure 3.5.
A linear symmetrical carbon dioxide molecule.
We can regard a carbon dioxide molecule as equivalent to a set of three particles
jointed by elastic springs (Fig. 3.5). Clearly the system will vibrate in some
manner in response to an external force. For simplicity we shall consider only
longitudinal vibrations, and the interactions of the oxygen molecules with one
another will be neglected, so we consider only nearest neighbor interaction. The
Lagrangian function L for the system is
L ˆ 12 m… _x2 1 ‡ _x 2 3†‡ 1 2 M _x2 2 1 2 k…x 2 x 1 † 2 1 2 k…x 3 x 2 † 2 ;
substituting this into Lagrange's equations
d @L
@L ˆ 0 …i ˆ 1; 2; 3†;
dt @ _x i @x i
we ®nd the equations of motion to be
x 1 ˆ k m …x 1 x 2 †ˆ k m x 1 ‡ k m x 2;
x 2 ˆ k M …x 2 x 1 † k M …x 2 x 3 †ˆ k
M x 1 2k
M x 2 ‡ k M x 3;
x 3 ˆ k
m x 2 k m x 3;
where the dots denote time derivatives. If we de®ne
0
B ~X ˆ @
x 1
x 2
x 3
0
k 1
m
C
A; ~A ˆ
k M
B
@
0
k
m
2k
M
k
m
0
k
M
k m
1
C
A
and, furthermore, if we de®ne the derivative of a matrix to be the matrix obtained
by di€erentiating each matrix element, then the above system of di€erential equations
can be written as
~X ˆ ~A ~X:
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