A.2 Interaction of classical oscillators of similar frequencies 301The determinantal equation yields
(
ω 12 −ω^2)(
ω 12 +2ω^22 −ω^2)
=0, (A.11)
giving the two eigenfrequenciesω=ω 1 andω′=
√
ω^21 +2ω^22 .For fre-
quencyω 1 the eigenvector isb=a, corresponding to a symmetric stretch
(Fig. A.1(b)); this is the same as the initial frequency sinceM 2 does not
move in this normal mode. The other normal mode of higher frequency
corresponds to an asymmetric stretch withb=−a(see Fig. A.1(c)). In
these motions the centre of mass does not accelerate, as can be verified
by summing eqns A.6, A.7 and A.8.
A treatment of two coupled oscillators of different frequencies is given
in Lyons (1998) in the chapter on normal modes; this is the classical
analogue of the non-degenerate case that was included in the general
treatment in the previous section. The books by Atkins (1983, 1994)
give a comprehensive description of molecules.