Homework Assignment 6
Physics 302, Classical Mechanics
Fall 2010
A. V. Kotwal
Handed out:
Friday, October 15, 2010
Due in class on:
Friday, October 22, 2010
Problems
1. The Lagrangian for a system of one degree of freedom can be written as
L
=
m
2
( ˙
q
2
sin
2
ωt
+ ˙
qqω
sin 2
ωt
+
q
2
ω
2
)
What is the corresponding Hamiltonian? Is it conserved? Introduce a new coordinate defined by
Q
=
q
sin
ωt
Find the Lagrangian in terms of the new coordinate, and the corresponding new Hamiltonian. Is the
new Hamiltonian conserved?
2. An object is bouncing vertically and perfectly elastically in an accelerating elevator.
If the time
dependence of the acceleration
a
(
t
) is slow enough to satisfy the adiabatic assumption, find the
maximum heights
h
max
(
t
) that the object reaches on its bounces [
h
max
(
t
) is measured relative to the
floor of the elevator].
3. A particle with mass
m
moves in one dimension under the influence of a potential
V
(
x
) =

k

x

,
where
k >
0. For energies that are negative, the motion is bounded and oscillatory.
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 Spring '09
 Physics, Angular Momentum, Energy, Force, Work, new coordinate, equivalent onedimensional problem

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