Linearization - Classical Mechanics - Lecture Slides, Slides of Classical Mechanics

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Linearization

Fixed

Point^ •

A

map

f

is

defined

on

a

metric

space

X

•^

Points

are

mapped

to

other

points

in

the

space.

•^

A

fixed

point

for

the

map

is

mapped

into

itself.

X

p

x^1

2

(^

x

x f^

f^

x^2

f

p

p f^

Hyperbolic

Fixed

Point

•^

The

Jacobian

is

a

partial

differential

matrix

n



m

matrix

•^

A

hyperbolic

matrix

has

no

unit

eigenvalues.

Unit

is

in

the

complex

plane

•^

A

hyperbolic

fixed

point

p

Smooth

map

f

on

R

n

Df

( p

)^

is

hyperbolic

•^

A

saddle

point

p

of

f

R

Real

eigenvalues



0 < |



| < 1 <|



|

j i

i j^

x

x f x f D x

Df

x

Mx

Topologically

Conjugate

•^

Let

F

,^

G

be

maps

F

is

a

map

on

space

X

G

is

a

map

on

space

Y

•^

If

there

exists

a

homeomorphism

h

h

:^

Y 

X

G

=

h

Fh

•^

Then

F

,^

G

are

topologically

conjugate.

h

X

Y

G

h

F

(^

y h F y G h

Eigenvalues

•^

The

origin

is

a

fixed

point.

•^

For

small

,^

there

are

two

approximate

solutions.

•^

The

generalized

variables

had

mass

and

length

folded

into

them.

2

ij V

2

2

ij G

^

12

g^ l

Eigenvalue

Results

•^

The

origin

is

a

fixed

point.

•^

For

small

x

,^

there

is

an

approximate

expansion

of

the

conjugate

map.

(^

x

Ax

x g

^0

Dg

lim

0



x x

x



A

Dg

lim

0



x

x

D

x



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