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Work - Analytical Mechanics - Lecture Slides, Slides of Applied Mechanics
In these Lecture slides, the Lecturer has discussed the following key concepts of Analytical Mechanics : Work, Generalized Coordinates, Transformation Rules, Changes, Generalized Velocity, Coordinates Themselves, Generalized, Generalized Force, Constraint Forces, Dependence
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2012/2013
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Work
Generalized
Coordinates
Transformation
rules
define
alternate
sets
of
coordinates.
3
N
Cartesian
coordinates
x
i
f
generalized
coordinates
q
m
Select
f
degrees
of
freedom
Small
changes
in
a
coordinate
can
be
expressed
by
the
chain
rule.
3
1
t
x
x
q
q
N
m
m
1
t
q
q
x
x
f
i
i
polar coordinate example
sin
cos
cos
cos
r
r
x
r
r
r
r
x
cos
sin
sin
sin
r
r
y
r
r
r
r
y
t t
x
q
q
x
x
i
f
m
m
m i
i
1
Generalized
Force
Force
acting
over
a
small
displacement
is
the
work.
Express
in
generalized
coordinates
Rewrite
the
work
in
terms
of
the
generalized
force
components,
Q
m
Q
t
Last
term
for
time
dependence
i
i
m
m
i m
i
t
t x
q
q
x
F
W
i
i
i
x
F
W
t x
F
Q
q
x
F
Q
t
Q
q
Q
W
i
i
i
t
i
m i
i
m
t
m
m
m
Constraint
Forces
All
the
Qm
are
applied
forces.
No
dependence
on
constraint
coordinates
Not
forces
of
constraint
Constraint
forces
do
no
work
Forces
of
constraint
are
often
unknown.
Newtonian
problem
complicated
by
them
i
m i
i
m
q
x
F
Q
Acceleration
and
Velocity
i
i
i
i
i
i
i
x x m x F W
)
(
The
work
can
be
expressed
by
mass
and
acceleration.
Mass
m
(
i
)
related
to
xi
The
Cartesian
coordinate
is
transformed
to
the
generalized
coordinate.
Use
the
boxed
identity
Work
expanded
in
terms
i
i
m
m
m i
i
i
t
t
x
q
q
x
x
m
W
)
(
i
i
i
m i
i
m
i m
i
i m
i
i
t
t x
x
m
q
q
x
dt d
x
q
x
x
dt
d
m
W
)
( ,
)
(
]
[
m i
i
m i
i
i m
i
q
x
dt d
x
q
x
x
q
x
x
dt d
Kinetic
Energy
The
velocity
can
be
used
to
find
the
kinetic
energy
T
Rearranging
summation
i
i
i
i
m
i
m
i m
i
m i
i
i
dt
t x
x
m
q
q
x
dt d
x
q
x
x
dt d
m
W
,
)
(
]
[
i
i
i
i
m
i
m i i i m i i i m
t
t x x m q x x m q x x m q
dt
d
W
)
(
,
)
(
1 2
)
(
1 2
)]
[
i
i
i
i
m
m
m
m
t
t x
x
m
q
q
T
q
T
dt d
W
)
(
]
[
Lagrangian
Function
Conservative
forces
depend
only
on
position.
Leave
non
‐
conservative
forces
on
the
right
side
of
the
equation
The
quantity
L
T
V
is
the
Lagrangian
This
gives
Lagrange’s
equations
of
motion.
For
f
equations,
2
f
constants
m
m
m
m
m
Q
q
V
Q
q
T
q
T
dt d
m
m
m
m
m
Q
q
V
q
T
q
V
dt d
q
T
dt d
m
m
m
Q
q
V
T
q
V
T
dt d
m
m
m
Q
q
L
q
L
dt d
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