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1 Homework #9
1. Consider a particle which is attracted to the origin with a force that de
pends only on the distance from the origin.
(a) Show that the Lagrangian for this motion has the form
L
(
X,V
) =
1
2
m

V

2

f
(

X

)
where
X
is the position,
V
is the velocity, and
f
is some function.
Solution
There is an error in the statement of this problem, so
reasonable discussions of kinetic and potential energies will constitute
correct answers.
(b) Examine the quantities energy, (linear) momentum, and angular mo
mentum. Which are conserved and which are not?
Solution
Since
H
=
L

V
·
L
V
=

1
2
m

V

2

f
(

X

) =

E,
the energy is constant in time, so energy is conserved.
Linear momentum is not conserved, because
∂L
∂
˙
x
i
=
m
˙
x
i
is not con
stant.
Angular momentum is conserved, which can be shown either by direct
computation as per the transformation on page 87 of the text, or by
noting that velocity and distance to the origin are independent of
rotation.
2. Consider the functional
F
(
u
) =
R
b
a
xu
0
(
x
)
2
dx
.
(a) Show
F
is invariant under
x
*
=
x,u
*
=
u
+
±
.
Solution
Find that
du
*
dx
*
=
d
(
u
+
±
)
dx
=
du
dx
a
*
=
a,b
*
=
b,
so
F
(
u
*
,I
*
) =
Z
b
*
a
*
x
*
±
du
*
dx
*
²
2
dx
*
=
Z
b
a
xu
0
(
x
)
2
dx
=
F
(
u
)
.
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 Spring '08
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