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Physics 325
Lecture 26
In several of the examples that we have done in the past two lectures, we have come upon
situations in which the derivative of the Lagrangian with respect to one of the generalized
coordinates is zero
0
i
L
q
∂
=
∂
(26.1)
Given the Lagrange equations
0
ii
LdL
qd
tq
∂
∂
−
=
∂∂
±
Equation (26.1) leads to the conservation law
0
i
dL
dt
q
∂
=
∂
±
(26.2)
or
constant
i
L
q
∂
=
∂
±
In Cartesian coordinates we have
LT
mx
p
xx
∂
∂
=
==
±
±±
Thus
0
i
L
x
∂
=
∂
leads to conservation of linear momentum along the
x
i
direction.
Now, in
Cartesian coordinates we also have that
i
LU
F
∂
∂
=
=−
which is the component of the force along the
x
i
direction.
In the situation where
Equation (26.1) is satisfied, then, we have
F
i
=0
and the component of the momentum
along the
x
i
direction is a constant in time.
This is good.
For the puck in a bowl problem from the last lecture, we found that the Lagrangian had
no explicit
φ
dependence (
0
L
∂∂=
), and this led to conservation of the
z
component of
angular momentum, as expected from the vanishing of the
z
component of the net torque
on the system.
Inspired by these discoveries, we define the
generalized momentum,
p
j
, associated with
the generalized coordinate
q
j
as
j
i
L
p
q
∂
≡
∂
±
and we note that
p
j
is conserved whenever
0
j
Lq
∂
∂=
.
In a conservative system the potential energy is independent of the velocity.
Therefore
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ii
UL
qq
i
T
q
∂
∂∂
==
∂
±±
±
We now defined the
generalized force
.
If the system is conservative we
have
FU
=−∂
∂
x
, so
ij
ji
j
j
xx
UU
FQ
qx
q
q
∂
∂
−
=
∂
∂
∑∑
−
where we have defined the generalized force as
i
i
i
i
x
U
Q
F
∂
∂
=−
=
∂
∂
∑
(26.3)
Note that if
q
i
is a length, then the partial derivative is dimensionless and
Q
i
is a force.
If
q
i
is an angle, then the partial derivative has the dimensions of length (
x
i
) and
Q
i
is a
torque (force x length).
Let’s see.
In polar coordinates we have
12
12 xy
yx
xc
o
s
x s
i
n
Q =F
F
F
F
sin
cos
Q
F
F
where
0
0
z
xy
xr
yr
xx xy
ry
r
x
k
FF
φ
φφ
ττ
∂∂ ∂∂
+=+
∂∂∂∂
=
=
=−=
=
GG
Systems with Holonomic Constraints
We consider a system of
n
particles (
3n
coordinates) with
m
equations of constraint:
( )
,
0
1,.
..,
kj
gqt
k
m
(26.4)
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 Fall '08
 Staff
 Physics, mechanics

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