Trajectories
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Trajectories - Analytical Mechanics - Lecture Slides, Slides of Applied Mechanics
In these Lecture slides, the Lecturer has discussed the following key concepts of Analytical Mechanics : Trajectories, Eulerian View, Described, Difficult, Properties, Density, Velocity, Fluid Change, Property, Derivative
Typology: Slides
2012/2013
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Trajectories
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Eulerian
View
In
the
Lagrangian
view
each
body
is
described
at
each
point
in
space.
Difficult
for
a
fluid
with
many
particles.
In
the
Eulerian
view
the
points
in
space
are
described.
Bulk
properties
of
density
and
velocity
0
t
r
r
t
r
t
r
v
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Compressibility
A
change
in
pressure
on
a
fluid
can
cause
deformation.
Compressibility
measures
the
relationship
between
volume
change
and
pressure.
Usually
expressed
as
a
bulk
modulus
B
Ideal
liquids
are
incompressible.
V
p
p
V
V
V
p
V
B
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Volume
Change
Consider
a
fixed
amount
of
fluid
in
a
volume
V
Cubic,
Cartesian
geometry
Dimensions
x
,
y
,
z
.
The
change
in
V
is
related
to
the
divergence.
Incompressible
fluids
must
have
no
velocity
divergence
z
z v
z
dt d
y
y v
y
dt d
x
x v
x
dt d
x y z
z
y
x
z v
y v
x v
V
dt d
z
y
x
V
v
V
dt d
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Transformation
Gradient
The
components
of
a
gradient
of
a
scalar
do
not
transform
like
a
position
vector.
Inverse
transformation
Covariant
behavior
Position
is
contravariant
Gradients
use
a
shorthand
index
notation.
i
i
,
i
i
x
e
m i
m
i
i
x q
q
x
m
m
q
e
m
m i
i
q
q
x
x
j i
i
j
x v
v
,
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Volume
Element
An
infinitessimal
volume
element
is
defined
by
coordinates.
dV
=
dx
1
dx
2
dx
3
Transform
a
volume
element
from
other
coordinates.
components
from
the
transformation
The
Jacobian
determinant
is
the
ratio
of
the
volume
elements.
1
1
1
q
x q
x
d
i
x
1
x
2
x
3
V d J q q q J V
q
q x
q
q x
q
x q
V
x
x
x
V
i
i
i
ijk
3
2
1
3 3 2 2 1 1
3
2
1
3
2
1
q
q
q
V
3
2
1
x
x
x
V
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Streamlines
A
streamline
follows
the
tangents
to
fluid
velocity.
Lagrangian
view
Dashed
lines
at
left
Stream
tube
follows
an
area
A
streakline
(blue)
shows
the
current
position
of
a
particle
starting
at
a
fixed
point.
A
pathline
(red)
tracks
an
individual
particle.
Wikimedia image
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Rotational
Flow
The
curl
of
velocity
measures
rotation
per
unit
area.
Stokes’
theorem
Fluid
with
zero
curl
is
irrotational.
Transform
to
rotating
system
with
zero
curl
Defines
angular
velocity
next
C
S
r
d
v
dS
v
n
3 2
v v
r
r
v
r
v
v
r
v
v
v
1 2
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