This preview shows pages 1–3. Sign up to view the full content.
Homework 12: # 10.13, 10.27, Cylinder
Michael Good
Nov 28, 2004
10.13
A particle moves in periodic motion in one dimension under the inﬂuence of a
potential
V
(
x
) =
F

x

, where
F
is a constant. Using actionangle variables, ﬁnd
the period of the motion as a function of the particle’s energy.
Solution:
Deﬁne the Hamiltonian of the particle
H
≡
E
=
p
2
2
m
+
F

q

Using the action variable deﬁnition, which is Goldstein’s (10.82):
J
=
I
p dq
we have
J
=
I
p
2
m
(
E

Fq
)
dq
For
F
is greater than zero, we have only the ﬁrst quadrant, integrated from
q
= 0 to
q
=
E/F
(where
p
= 0). Multiply this by 4 for all of phase space and
our action variable
J
becomes
J
= 4
Z
E/F
0
√
2
m
p
E

Fq dq
A lovely usubstitution helps out nicely here.
u
=
E

Fq
→
du
=

F dq
J
= 4
Z
0
E
√
2
mu
1
/
2
1

F
du
J
=
4
√
2
m
F
Z
E
0
u
1
/
2
du
=
8
√
2
m
3
F
E
3
/
2
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Goldstein’s (10.95) may help us remember that
∂H
∂J
=
ν
and because
E
=
H
and
τ
= 1
/ν
,
τ
=
∂J
∂E
This is
τ
=
∂
∂E
[
8
√
2
m
3
F
E
3
/
2
]
And our period is
τ
=
4
√
2
mE
F
10.27
Describe the phenomenon of small radial oscillations about steady circular mo
tion in a central force potential as a onedimensional problem in the actionangle
formalism. With a suitable Taylor series expansion of the potential, ﬁnd the pe
riod of the small oscillations. Express the motion in terms of
J
and its conjugate
angle variable.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 03/17/2012 for the course PHYS 202 taught by Professor Atkin during the Spring '12 term at Amity University.
 Spring '12
 atkin
 Work

Click to edit the document details