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Physics Stability Analysis: Lagrangian Equilibria, Matrix Stability, Orbital Potentials, Slides of Classical Mechanics
Stability analysis in physics through various concepts such as lagrangian equilibria, matrix stability, orbital potentials, and lyapunov stability. It covers topics like expanding lagrangians near equilibrium, second derivative tests, normal modes, and effective potentials. Useful for students in physics and related fields.
Typology: Slides
2012/2013
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Stability
Lagrangian
Near
Equilibium
•^
A
‐dimensional
Lagrangian
can
be
expanded
near
equilibrium.
Expand
to
second
order
2 , 2 2 , 2 2 , 2 2 , , , 0 0 0 0 0 0 0 0 0 0 0 0
q
L q
q q
q q
L
q
L q
q
L q
q
L q
L
L
q q q q q q q q q q q
q^
^
^
2
2
q
F
q
Eq
Dq
q
C
Bq
A
L
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Matrix
Stability
•^
A
general
set
of
coordinates
gives
rise
to
a
matrix
form
of
the
Lagrangian.
Normal
modes
for
normal
coordinates.
•^
The
eigenfrequencies
2
determine
stability.
If
stable,
all
positive
Diagonalization
of
V
j i ij
j i ij^
q q V q q G L
1 2
1 2
] ) ( ) ( )
[(
2
2
2
1 2
j
j
j
i^
x
q
c
L
(^
2
j
Orbital
Potentials
•^
Kepler
orbits
involve
a
moving
system.
Effective
potential
reduces
to
a
single
variable
Second
variable
is
cyclic
2 2 2
r J
k r
V
eff
dr dV
r
eff
r
r
V
eff
r^0
r^
r^0
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Modified
Kepler
•^
Kepler
orbits
can
have
a
perturbed
potential.
Not
small
at
small
r
Two
equilibrium
points
Test
with
second
derivative
Test
with
r
2 2
3
r J
b r
k r
V
eff
b k
k J
k J
r^
2 (^42)
2
0
r
V
eff
r^0
r A
2 3
4
2
J r
r b
k r
dr dV
eff
^0
2
2
b k
r k J
r
b k
k J
k J
rA
2 (^42)
2
stable unstable
Lyapunov
Stability
•^
A
Lyapunov
function
is
defined
on
some
region
of
a
space
X
including
Continuous,
real
function
•^
The
derivative
with
respect
to
a
map
f
is
defined
as
a
dot
product.
•^
If
V
exists
such
that
V
,^
then
the
point
is
stable.
(^
x
x V
*^
x f x V x V