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Lecture 14: Small oscillations (5 Oct 09)
A. Small amplitude expansion
1. FW Sec. 21 starts with 1D picture (expand about a position of zero
force) and then to quadratic forms for expansion of both kinetic energy
and potential energy:
T
=
X
j
m
j
2
˙
x
2
j
=
X
j
m
j
2
[
X
i
∂x
j
∂q
i
˙
q
i
]
2
≡
1
2
X
ij
T
ij
˙
q
i
˙
q
j
where
T
ij
=
T
ji
(if not seen immediately, juggle the dummy indices) [
T
must be a positive deﬁnite matrix, from its role in the (positive) kinetic
energy]. Then with the
q
i
being increments relative to a position with
zero gradients:
ΔΦ =
1
2
X
ij
Φ
ij
q
i
q
j
The matrix Φ
ij
must be positive deﬁnite [
x
T
·
Φ
·
x
>
0] for stable
oscillations but that is not automatic for expansion about an arbitrary
point.
2. Cautionary remark for molecular vibrations (e.g., triatomic molecule).
Expand the potential Φ(
r
ij
) relative to its equilibrium length
r
0
using
r
i
j
=
r
0
ˆ
n
+
δ
r
ij
:
r
ij
’
r
0
+ ˆ
n
·
δ
r
ij
Φ(
r
ij
)
’
Φ(
r
0
) +
1
2
[ˆ
n
·
δ

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