1
CDS 140a: Homework Set 7
Due: Wednesday, December 2, 2009.
For the first three questions, consider the mechanical system shown in the Figure
below. The mass
M
slides (without friction) along the curve
y
=
x
2
. The mass
m
hangs by a light rod of length
l
(as a planar pendulum) from the mass
M
. Let
θ
denote the angle with the vertical, as shown. Both masses are subject to a downward
gravitational force.
M
m
y
x
g
; gravity
θ
y
=
x
2
1. Write down a Lagrangian for the system and compute the Euler–Lagrange
equations.
2. Does this system have a conserved energy? Explain how to get this expression
in both the Lagrangian and the Hamiltonian formulations of the problem.
3. Note that when
M
is stationary at
x
= 0 and
m
is hanging vertically (with
θ
= 0), one has an equilibrium. Is it stable? Asymptotically stable? What if
one adds friction?
4. The support point of a simple planar pendulum of mass
m
and length
l
moves
along the horizontal
x
axis with position
x
(
t
) =
a
cos
ωt
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Marsden
 Mass, Dynamical systems, downward gravitational force, planar pendulum, simple planar pendulum

Click to edit the document details