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Goldstein 119 (3
rd
ed. 1.21)
Start with a Cartesian coordinate system with the origin at the hole.
In
these coordinates, the kinetic energy is
T
=
1
2
m
1
(
·
x
2
+
·
y
2
)+
1
2
m
2
·
z
2
,
(1)
where
m
1
and
m
2
are the particle masses,
x
and
y
determine the position of
particle 1 in the plane, and
z
determines the position of particle 2 below the hole.
The system has only two degrees of freedom because the position of particle 2
on the
z
axis is constrained by the string length
s
. Use the polar coordinates
r
and
θ
as the two generalized coordinates for this system,
x
=
r
cos
θ
,
(2)
y
=
r
sin
θ
,
(3)
and the constraint is
z
=
r
−
s.
(4)
In terms of
r
and
θ
, we have the following expression for
T
T
=
1
2
(
m
1
+
m
2
)
·
r
2
+
1
2
m
1
(
r
·
θ
)
2
,
(5)
and the potential energy is
V
=
m
2
gz
=
m
2
g
(
r
−
s
)
.
(6)
It follows that the Lagrangian,
L
=
T
−
V
,is
L
=
1
2
m
·
r
2
+
1
2
m
1
(
r
·
θ
)
2
−
m
2
g
(
r
−
s
)
,
(7)
where
m
is the total mass
m
=
m
1
+
m
2
.
(8)
1
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 Fall '10
 Wilemski
 mechanics, Energy, Kinetic Energy, Mass

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