Goldstein Problem 2.11 (3
rd
ed. # 13)
While the motion of the mass point is constrained to follow the hoop, the
system has only one degree of freedom. To calculate the reaction of the hoop
on the particle, we treat the system as having two degrees of freedom and use a
Lagrange multiplier
λ
to enforce the constraint. Use plane polar coordinates as
the two generalized coordinates. The polar angle
θ
will be measured from a
y
axis that points upwards so that
θ
is positive when the mass point is displaced
away from its initial position.
The other generalized coordinate will be the
radial displacement
r
of the mass from the center of the hoop.
The kinetic energy of the mass point is
T
=
m
2
(
·
x
2
+
·
y
2
)
.
(1)
In the polar coordinate system we have
x
=
r
sin
θ
,
(2)
y
=
r
cos
θ
,
(3)
from which it follows that
·
x
=
·
r
sin
θ
+
r
·
θ
cos
θ
,
(4)
·
y
=
·
r
cos
θ
−
r
·
θ
sin
θ
,
(5)
and Eq.(1) can then be written as
T
=
1
2
m
[
·
r
2
+
r
2
·
θ
2
]
.
(6)
The potential energy may be taken as
V
=
mgy
=
mgr
cos
θ
.
(7)
With Eqs.(6) and (7), the Lagrangian for this system is
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 Fall '10
 Wilemski
 mechanics, Mass, Cos, Coordinate system, Polar coordinate system, mg cos

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