Physics 325
Spring 2011
Homework
7A
Name: ____________________________
Due:
Apr 6, 2011 by 9am
(lecture or 325 box)
1.
(10 points)
A soap bubble placed between two centered hoops will take the shape with the
minimum area. Use the calculus of variations to find the curve
( )
y x
that defines the surface of
revolution.
Hint: Assume that the surface is circular everywhere in the
x
z
plane. The problem then
reduces to a twodimensional problem with an element of surface area written as
2
xds
, where
ds
is an element of length in the
x
y
plane.
2.
(10 points)
In relativity theory, velocities can be represented by points in a certain “rapidity
space” in which the distance between two neighboring point is
2
2
2
2
2
1
ds
dr
r d
r
where
r
and
are polar coordinates, and we consider just a twodimensional space. (An
expression like this for the distance in nonEuclidean space is often referred to as the
metric
of
the space.) Use the EulerLagrange equation to show that the shortest path from the origin to any
other point is a straight line.
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 Spring '08
 Staff
 mechanics, Energy, Work, Coordinate system, Lagrangian mechanics

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