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
.
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- Fall '09
- Marsden
- Mass, Dynamical systems, downward gravitational force, planar pendulum, simple planar pendulum
-
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