Homework Assignment 5
Physics 302, Classical Mechanics
Fall, 2010
A. V. Kotwal
Handed out:
Friday, October 1, 2010
Due in class on:
Friday, October 8, 2010
Problems
1. Consider the Hamiltonian of a onedimensional simple harmonic oscillator,
H
(
q, p
) =
1
2
p
2
+
1
2
ω
2
q
2
.
(a) Use Hamilton’s equations of motion to solve for
q
(
t
) and
p
(
t
) in terms of initial values (
q
0
, p
0
).
(b) Evaluate the Poisson bracket
{
q
(
t
)
, p
(
t
)
}
with respect to (
q
0
, p
0
).
(c) Show that the relationship between (
q
(
t
)
, p
(
t
)) and (
q
0
, p
0
) represent a canonical transformation.
2. Show that the function
S
(
q, P, t
) =
mω
2
(
q
2
+
P
2
) cot
ωt

mωq P
csc
ωt
is a solution of the HamiltonJacobi for Hamilton’s principal function for the simple harmonic oscil
lator with
H
=
1
2
m
(
p
2
+
m
2
ω
2
q
2
)
.
Show that this function generates a correct solution to the motion of the harmonic oscillator.
3. Suppose the potential in a problem of one degree of freedom is linearly dependent upon time, such
that the Hamiltonian has the form
H
(
x, p, t
) =
p
2
2
m

m A t x,
where
A
is a constant.
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 Spring '09
 Physics, Energy, Work, Hamiltonian mechanics, Simple Harmonic Oscillator

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