Action-Angle - Classical Mechanics - Lecture Slides, Slides of Classical Mechanics

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Action

Angle

Dimensional

CT

•^

Define

a

‐D

generator

S’.

Time

‐independent

H

.

Require

new

conjugate

variables

to

be

constants

of

motion.

•^

Conjugate

momentum

is

a

constant

J

Hamiltonian

is

constant

Conjugate

position

is

cyclic

Linear

in

time

(^

t p q p q S j

j

  

S^ q

p

S p

q

t

S J

w

 

  

H J

 w

(^

p q H

E p q H J H 

J

p

a frequency

 is a constant, ie



from HJ



units of action

Alternate

Generators

Generating

functions

differ

by

a

Legendre

transformation.

The

transformation

can

be

expressed

as

type

I.

S

is

also

periodic

with

period

(^

J q S p q S

 

(^

w q S q q S

Jw

S

q p

S

S

S^ q

p

S w

J^

Simple

Oscillator

•^

The

oscillator

H

is

constant

and

expressed

in

terms

of

p

•^

The

action

can

be

integrated

•^

The

generator

can

be

defined

from

the

action

^

dq

q

E

J

(^2) / 1 2 2

E

q

p

H

2 2

2



 



 

E

E

J

2

^

^  

^

dq

q

J

J

q S

(^2) / 1 2 2 ) , (



 

Physical

View

) , ( ) , (^

J q S p q S^

 

^

•^

The

motion

in

phase

space

is

harmonic.

Amplitudes

of

q, p

Area

in

phase

space

is

the



times

the

action.

Angle

w

repeats

per

cycle.

2 a

J

E

a

p

q

J = E/



^

w

 =



 t

w

a

q

 2 sin



w

a

p





cos 2

Generating

Function

The

generating

function

S’

can

be

found

by

integration

and

substitution.

The

function

S

comes

from

the

Legendre

transformation

w w q w q w q S

csc

cot

(^

2

2

2

Jw

w

q

w q S

cot

(^

2

Jw

S

q p

S

S

w

q

S

cot

2