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GPS 3.33
We can use a cylindrical coordinate system (
ρ
,
θ
,
z
) to describe the position
of the particle.
The polar (
z
) axis points upwards,
ρ
is the radial distance
measured from the
z
axis (
ρ
=
p
x
2
+
y
2
), and
θ
is measured in the
x

y
plane
from the
x
axis. Because the particle is constrained to move on the surface of a
paraboloid of revolution, this system has only two degrees of freedom. We will
use
ρ
and
θ
as generalized coordinates to set up the Lagrangian after eliminating
z
and
·
z
with the holonomic equation of constraint. In cylindrical coordinates
the kinetic energy of a single particle is
T
=
m
2
(
·
ρ
2
+
ρ
2
·
θ
2
+
·
z
2
)
.
(1)
The potential energy for this system is
V
=
mgz ,
(2)
and the constraint equation is
z
=
aρ
2
,
(3)
where
a
is a constant that determines the shape of the paraboloid of revolution.
From Eq.(3), it follows that
·
z
=2
aρ
·
ρ.
(4)
From these four equations we may write the Lagrangian in terms of
ρ
,
·
ρ
,and
·
θ
as
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.
 Fall '10
 Wilemski
 mechanics

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