Physics 409/Classical Mechanics
Test #6
7 December 2010
Name
ANSWERS
Answer all parts of each of the two questions.
50
points total. The points value of each part is
indicated.
Begin the answer to each question on a new sheet of paper.
Clearly define all
coordinates and variables. Be sure to include sufficient detail in each answer so that your logic is
clear to me.
(/
)
ii
i
hq
L
q
L
=∂
∂
−
∑
±±
(1)
(25 pts.)
A simple pendulum (mass
m
at the end of massless rod of length
l
) is attached to a
cart constrained to move at constant speed v
0
in the
x
direction. The pendulum oscillates in the
x

y
plane only, where
y
is the vertical direction (the direction of
gravity).
Assume
y
= 0 where the pendulum attaches to the
horizontal support bar.
(a) (13 pts.)
Using appropriate coordinates set up a correct
Hamiltonian for the system. (The Lagrangian and energy
function
h
should be explicitly derived.)
(b) (4 pts.)
Ignoring any additive constant terms, what
physical quantity does your Hamiltonian represent?
Is your
Hamiltonian conserved?
(c) (4 pts.)
Find the Hamiltonian equations of motion for the system.
(d) (4 pts.)
Combine the Hamiltonian equations of motion and show that they reduce to a single
EOM for a simple pendulum.
ANS:
Let
x
= v
0
t
+
l
sin
θ
and
y =
!
l
cos
θ
. The Lagrangian is
22
00
(
/ 2)[(
)
2 v
v ]
Lm
l
l
θθ
=+
+
and with
we find
cos
mgl
θ
+
0
/(
v
c
o
s
)
pL
m
l
l
∂ =
+
0
(
/ 2)[(
)
v ]
hm
l
=−
±
.
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 Fall '10
 Wilemski
 mechanics, Energy, mgl cos

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