1.685J/2.0345/18.377J Nonlinear Dynamics and Waves
Spring 2007
Problem Set No.
1
Out:
Thursday, February 15, 2007
Due:
Thursday, March
1, 2007 in class
Problem 1
A man rows a boat across a river of width
a occupying the region 0
5
x
5
a in the x, y
plane, always rowing towards a fixed point on one bank, say (0,O). He rows at constant
speed
u
relative to the water, and the river flows at constant speed
v.
Show that
where (x, y) are the coordinates of the boat. Show that the phase trajectories are given
where
a
=
vlu.
Sketch the phase diagram for
a
<
is the origin? What happens to the boat if
a
>
I?
Problem 2
Consider the system
Discuss the stability of the equilibrium point at the origin (i) by linearized thcory and (ii)
for the full nonlinear system. (Hint: consider x4
+
2y2)
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3
The equation of motion of a bar restrained by springs and attracted by a parallel current
carrying conductor is
where
c,a
and
X
are positive constants. Sketch the phase trajectories for
xo
<
x
<
a
and classify all equilibrium points for
X
>
0.
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 Spring '07
 TriantaphyllosAkylas
 Economics, Consumer spending, Equilibrium point, Nonlinear system, Stability theory

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