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) Stephenson-Kapitza 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 Stephenson-Kapitza pendulum, choosing as the co-
ordinate the angular displacement
θ
away from the downward vertical.
b) Hence, obtain the equation of motion for the Stephenson-Kapitza pendulum.
What is striking about the Stephenson-Kapitza 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), 755-768; and/or
http://www.fam.web.mek.dtu.dk/FVP/Jon/Theory1.html
. The method of multiple-scale
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,