Handout 5
3 September 2010
Quantum Mechanics I
webusers.physics.illinois.edu/
∼
goldbart/
Homework 1
Prof. P. M. Goldbart
University of Illinois
This homework is due to be handed in to the PHYS 580 Homework Box by 2:30 p.m. on
Thursday September 9.
1) Lagrangian for a compound plane pendulum
: Consider a pendulum comprising an
upper segment, of length
L
1
and bob mass
M
1
, and a lower segment that is freely hinged
to the bob of the upper segment, of length
L
2
and bob mass
M
2
. The segments move
in a common vertical plane in the presence of gravity (acceleration
g
). Let the angular
displacements of the upper and lower segments from the vertical direction be, respectively,
θ
1
and
θ
2
.
a) Construct a Lagrangian for the compound plane pendulum in terms of
θ
1
and
θ
2
.
b) Hence, obtain the equations of motion for the compound plane pendulum.
c) By linearizing about the position of mechanical equilibrium, determine the frequen
cies of the normal modes of oscillation of the compound pendulum in terms of its
parameters,
2) StephensonKapitza pendulum
: This type of pendulum is a simple pendulum of
length
L
and natural frequency
ω
, except that the point of support is forced to vibrate. In
the simplest case, which we consider here, the point of support undergoes vertical harmonic
oscillations of frequency Ω
ω
and amplitude
qL
.
a) Construct the Lagrangian for the StephensonKapitza pendulum, choosing as the co
ordinate the angular displacement
θ
away from the downward vertical.
b) Hence, obtain the equation of motion for the StephensonKapitza pendulum.
What is striking about the StephensonKapitza pendulum is that if the support is vibrated
suﬃciently rapidly (i.e., for
q
Ω
>
√
2), the pendulum has a stable conﬁguration when it
is inverted (i.e., when
θ
=
π
), about which it can undergo stable, small oscillations at
a frequency given by
ω
√
−
1 +
1
2
q
2
Ω
2
. For further details, see E.I. Butikov,
On the dy
namic stabilization of an inverted pendulum
, Am. J. Phys.
69
(2001), 755768; and/or
http://www.fam.web.mek.dtu.dk/FVP/Jon/Theory1.html
. The method of multiplescale
analysis provides a mathematically precise approach to this problem; for this and many
other powerful techniques, see, e.g., C.M. Bender and S.A. Orszag,
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 Fall '10
 Staff
 mechanics, Work, Fundamental physics concepts, Lagrangian mechanics, compound plane pendulum, ϵµνρ ϵµνρ

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