Physics 498/MMA
Handout FT
Dec 20th 2002
Mathematical Methods in Physics I
http://w3.physics.uiuc.edu/
∼
mstone5
Prof. M. Stone
2117 ESB
University of Illinois
This exam has
four
pages and
six
problems. Answer question
one
, and then any
other
three
questions.
Do not hand in solutions to more than this number of problems!
Try to answer entire questions. Little, if any, credit will be given for fragmentary answers.
Errors will not be propagated, so make sure of each step before you go on.
1) Onedimensional Green Function
: Consider the differential equation

y
00
=
f
(
x
)
with boundary conditions
y
0
(0) = 0 and
y
(1) = 0. We are given
f
(
x
) and wish to solve for
y
(
x
).
a) Construct the explicit Green function appropriate to this problem (5 points).
b) Use your Green function to write down the solution of the boundary value problem as
the sum of two explicit integrals over complementary components of the unit interval
(5 points).
c) Evaluate the
x
derivative of your solution,
y
(
x
), and confirm that
y
obeys both bound
ary conditions (5 points).
d) Take one further derivative of your
y
(
x
) and confirm that it does indeed solve the
original problem (5 points).
2) Bead and string
: A bead of mass
M
is free to slide up and down the
y
axis.
x
y
y(0)
0
L
A bead connected to a string.
It is attached to the
x
= 0 end of a string in such a way that the Lagrangian for the
stringbead system is
L
=
1
2
M
[ ˙
y
(0)]
2
+
Z
L
0
1
2
ρ
˙
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 Fall '07
 Stone
 Boundary value problem, Green function

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