PHYS 3202 Classical Mechanics II - Homework #7
Due at 12:05pm, Friday March 18, 2011 in the class
Please show intermediate steps of your calculations. Whenever appro-
priate, you may draw diagram(s).
Problem 1
(5 points): a particle undergoing periodic motion in a conservative
force field traces out a symplectic area
A
=
Z
τ
0
p
·
d
r
in each period
τ
. The particle’s kinetic energy is
T
. If the particle’s speed is much
smaller than the speed of light in vacuum, show that
A
= 2
h
T
i
τ,
where
h
T
i
is the average kinetic energy in each period:
h
T
i ≡
1
τ
R
τ
0
Tdt
. (To prevent
confusion with the kinetic energy, we use symbol
τ
for the period.)
Problem 2
(15 points): For each of the following 3 systems, compute the energy
E
as a function of the symplectic area
A
=
R
τ
0
p
·
d
r
within each period
τ
, and check
whether
E
∝
A
β
(if yes, give the value of
β
).
For each system, also compute the
frequency of the motion
ν
≡
1
/τ
, and then verify the relation
ν
=
E
0
(
A
).
Omit
gravity.
For system 2, please eliminate the radius
r
in favor of
A
in your
results.

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
This is the end of the preview.
Sign up
to
access the rest of the document.