This preview shows pages 1–3. Sign up to view the full content.
Lecture 15: Oscillators II (7 Oct 09)
review 5 Oct HW
A. Review
1. Expansion of both kinetic energy and potential energy in departures
from equilibrium positions
x
(0)
j
, so derivatives are evaluated at
q
(0)
j
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
ΔΦ =
1
2
X
ij
Φ
ij
q
i
q
j
where
T
and Φ
ij
are real symmetric matrices,
T
ij
=
T
ji
and Φ
ij
=
Φ
ji
. Because the kinetic energy is intrinsically a positive quantity, the
matrix
T
ij
must be positive deﬁnite. Also Φ
ij
must be positive deﬁnite
for stable oscillations, but that is not automatic for arbitrary expansion
point.
2. The Lagrange equations then become:
X
j
T
ij
¨
q
j
=

X
j
Φ
ij
q
j
3. Look for oscillator solutions in form
δq
j
=
A
j
exp(

ıωt
+
φ
j
) =
q
0
j
exp(

ıωt
)
4. Then the Lagrange equations are
ω
2
X
j
T
ij
q
0
j
=
X
j
Φ
ij
q
0
j
5. This has the form of a generalized eigenvalue problem
λ
T
·
q
=
M
·
q
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentTheorems that
λ
and the eigenvectors are real carry over to this form,
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09
 BRUCH
 Energy, Kinetic Energy, Potential Energy

Click to edit the document details