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Physics 325
Lecture 20
Generalized Forces and Momenta
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
(20.1)
Given the Lagrange equations
0
ii
L
d
L
q
dt q
Equation (20.1) leads to the conservation law
0
i
dL
dt q
(20.2)
or
constant
i
L
q
In Cartesian coordinates we have
i
i
LT
xx
mx
p
Thus
0
i
L
x
leads to conservation of linear momentum along the
i
x
direction.
Now, in
Cartesian coordinates we also have
i
LU
F
which is the component of the force along the
x
i
direction.
In the situation where Equation (20.1) is satisfied, then, we have
0
i
F
and the
component of the momentum along the
i
x
direction is a constant in time. This is good.
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View Full Document For the puck in a bowl problem from the last lecture, we found that the Lagrangian had
no explicit
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,
j
p
, associated with
the generalized coordinate
q
i
as
j
j
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
0
and
j
j
j
U
L
T
q
q
q
We now defined the
generalized force
.
If the system is conservative, we have
F
i
U
x
i
, so
i
i
j
i
j
j
i
i
i
j
x
UU
q
x
q
Q
x
F
q
where we have defined the generalized force as
j
j
i
i
i
j
U
Q
q
x
F
q
(20.3)
Note that if
j
q
is a length, then the partial derivative is dimensionless and
j
Q
is a force.
If
q
j
is an angle, then the partial derivative has the dimensions of length (
x
j
) and
Q
j
is a
torque (force
length).
Let’s see.
In polar coordinates we have
12
xy
yx
x
cos
x
sin
Q = F
F
FF
sin
cos
Q
F
F
where
0
0
z
x
r
y
r
xx
r
y
r
x
i
j
k
Systems with Holonomic Constraints
We consider a system of
n
particles (
3n
coordinates) with
m
equations of constraint:
,
0
1,.
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff
 mechanics, Force

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