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Phase Space and Dynamics: Phase Curves, Portraits, Flow, and Liouville's Theorem, Slides of Applied Mechanics
An overview of phase space concepts in physics, including phase curves, portraits, flow, and liouville's theorem. It covers undamped and damped harmonic motion, phase portraits for lagrangian and hamiltonian systems, and the concept of phase flow. The document also explains the significance of liouville's theorem in conserving the phase space density.
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Phase
Space
Phase
Curve
‐D
Harmonic
motion
can
be
plotted
as
velocity
vs
position.
Momentum
instead
of
velocity
For
one
set
of
initial
conditions
there
is
a
phase
curve.
Ellipse
for
simple
harmonic
Spiral
for
damped
harmonic.
q
p
q
,
UndampedDamped
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Phase
Portrait
•^
A
series
of
phase
curves
corresponding
to
different
energies
make
up
a
phase
portrait.
Velocity
for
Lagrangian
system
Momentum
for
Hamiltonian
system
q
p
q
,
^ E
< 2
E
E = 2
> 2
Phase
Flow
A
region
of
phase
space
will
evolve
over
time.
Large
set
of
points
Consider
conservative
system
The
region
can
be
characterized
by
a
phase
space
density.
dV
N
t^2
t^
t^1
t^
q
p
^
j
j
j^
dp
dq
dV
Differential
Flow
The
change
in
phase
space
can
be
viewed
from
the
flow.
Flow
in
Flow
out
Sum
the
net
flow
over
all
variables.
j
j
j
j
in
q
dp dt
p
dq dt
^
^
^
^
j j
j j
j j
j j^
p q
p p
q q
p q t
^
^
^
^
j
j
j j
j
j
j
j j
j out
q
p
p p
p
p
q
q q
q
0
^
j^
j j
j j
j j
j j^
p p
p p
q q
q q
t
q
p
j q
j p j p
j q
Liouville’s
Theorem
Hamilton’s
equations
can
be
combined.
Simplify
phase
space
expression
This
gives
the
total
time
derivative
of
the
phase
space
density.
Conserved
over
time
j
j
p
H q
j
j
q
H p
0
j j
j j
q q
p p
0
^
j
j j
j j
p p
q q
t^
0
d^ ^ dt