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Mechanics

Physics 151

Lecture 20

Canonical Transformations

(Chapter 9)

What We Did Last Time „

Hamilton’s Principle in the Hamiltonian formalism^ „

Derivation was simple „

Additional end-point constraints

„

Not strictly needed, but adds flexibility to the definitionof the action integral

„

This connects to: Canonical Transformations

„

Principle of Least Action

1

2

1

2

(^

)^

(^

)^

q t

q t

p t

p t

(^ =

)

(^21)

t

i^

i

t

I^

p q

H q p t

dt

δ

δ ≡

∫^

(^21)

t

i^

i

t^

p q dt

∫^

^

Got intothis a bit

General Transformation „

Two types of transformations are possible^ „ „ „

Both satisfy Hamilton’s principle

„

Combined, we find

(^

)

(^21)

t

i^

i

t^

p q

H q p t

dt

δ

∫

^

(^

)

(^21)

(^

,^

t

i^

i

t^

PQ

K Q P t

dt

δ

∫

and

(^

i^

i^

i^

i

PQ

K

p q

H

^

i^

i^

i^

i

dF

PQ

K

p q

H

dt

^

Scale transformationCanonical transformation

(^

i^

i^

i^

i

dF

PQ

K

p q

H

dt

λ

^

^

Extended Canonical

transformation

Scale Transformation

„

We can always change the scale of (or unit we use tomeasure) coordinates and momenta „

To satisfy Hamilton’s principle, we can define „

This is trivial

„

We now concentrate on – Canonical transformations

(^

,^

(^

,^

K P Q t

H

p q t

μ

ν

=

i^

i

P

p

i^

i

Q

q

(^

i^

i^

i^

i

PQ

K

p q

H

^

^

Scale transformation

Simple Example [1] „

Try a generating function:^ „

Canonical transformation generated by

F

is

„

OK, that was too simple

„

Let’s push this one step further…

i^

i^

i^

i

F

q P

Q P

(^

i^

i^

i^

i^

i^

i^

i^

i^

i

dF

PQ

K

K

q

Q

P

Pq

p q

H

dt

^

^

^

i^

i

Q

q

i^

i

P

p

Identity transformation

K

H

i^

i^

i^

i

dF

PQ

K

p q

H

dt

^

Simple Example [2] „

Let’s try this one:^ „

fi

are arbitrary functions of

q

q

n^

and

t

„

We can do all what we could do before

(^

)^

i^

i

i^

i^

i^

i^

i^

i^

j^

i^

i^

i

j f^

f

dF

PQ

K

K

f^

Q

P

P

q

P

p q

H

dt

q

t

^

^

^

1 (^

,^

,^

i^

n^

i^

i^

i

F

f^

q

q

t P

Q P

1 (^

,^

,^

i^

i^

n

Q

f^

q

q

t

j

i^

j i f

p

P

q ∂ =

All “point transformations” of generalized coordinates are covered

i

i

f

K

H

P

t ∂

Must invert these

n

equations to get

P

i

i^

i^

i^

i

dF

PQ

K

p q

H

dt

^

Finding the Generator „

Let’s look for a generating function^ „

Suppose

for simplicity

„

Easiest way to satisfy this would be

„

Trivial example:

i^

i^

i^

i

dF

PQ

K

p q

H

dt

−

=

−

^



(^

,^

K Q P t

H q p t

i^

i^

i^

i

dF

p q

PQ

dt

F

F q Q

i

i F

P

∂^ Q

i

i F

p

q ∂

)^

i^

i

F q Q

q Q

i^

i

p

Q

i^

i

P

q

=

In the Hamiltonian formalism,

you can freely swap the coordinates and the momenta

Type-1 Generator „

is not very general

„

It does not allow

t

-dependent transformation

„

Fix this by extending to „

This affects the Hamiltonian

F

F q Q

F

F q Q t

Call it Type-

i

i

F q Q t

P

Q

i

i

F q Q t

p

q

i^

i^

i^

i

dF

PQ

K

p q

H

dt

−

=

−

^



1

1

1

i^

i^

i^

i^

i^

i

i^

i

F

F

F

dF

q

Q

p q

PQ

K

H

dt

q

Q

t

^

^

1 F

K

H

t ∂

Harmonic Oscillator „

Let’s try a Type-1 generator^ „

Express

p

as a function of

q

and

Q

„

Integrate with

q

(^

) cos

p

f^

P

Q

(^

sin

f^

P

q

Q

m

F q Q t

1 F

p

q ∂ =

1 F

P

Q

cot

p

m

q

Q

ω

=

2

1

cot

2 m

q

F

Q

2

1

cot

m

q

F

Q

1

2

2sin

F

m

q

P

Q

Q

We are getting somewhere

Harmonic Oscillator

„

We need to turn

H

(q

,^

p

) into

K

(Q

,^

P

„

Solve the above equations for

q

and

p

„

Now work out the Hamiltonian

„

Things don’t get much simpler than this…

1

cot

F

p

m

q

Q

q

2

1

2

2sin

F

m

q

P

Q

Q

sin P

q

Q

m

cos

p

Pm

Q

(^

)

2

2

2

2

K

H

p

m

q

P

m

„

Oscillator moves in the

p-q

and

P-Q

phase spaces

„

One cycle draws the same area

in both spaces

„

The area swept by a cyclic system in the phase space isinvariant

Phase Space

p

q

2 2 E m^ ω

2 mE

Q

P E^ ω

E

π^ ω

Will come back to this in Lecture 23

Other Types of Generators „

Type-1 generator

is still not so general

„

Just try to find a generator for

„

We need generating functions of different set ofindependent variables^ „

In fact, we may have 4 basic types of them

„

We can derive them using the now-familiar rule^ „

i.e. we can add any

dF/dt

inside the action integral ( , 1

F

F q Q t

i^

i

Q

q

i^

i

P

p

F q Q t

2

F

q P t

,^

F

p Q t

4

(^

,^

F

p P t

Type-2 Generator „

If we go back to the original definition of generatingfunction^ „

Trivial case:

„

We push the same idea to define the other 2 types

i^

i^

i^

i

dF

PQ

K

p q

H

dt

^

2

i^

i

F

F

q P t

Q P

2

i

i F

Q

P

2

i

i F

p

q ∂

2 F

K

H

t ∂

2

i^

i

F

q P

i^

i

Q

q

i^

i

p

P

Identity transformation

Four Basic Generators

Trivial Case

Derivatives

Generator

F q Q t

2

i^

i

F

q P t

Q P

,^

i^

i

F

p Q t

q p

4

(^

,^

i^

i^

i^

i

F

p P t

q p

Q P

1

i

i F

p

q ∂ =

1

i

i F

P

Q

1

i^

i

F

q Q

i^

i

Q

p

i^

i

P

q

=

2

i

i F

p

q ∂ =

2

i

i F

Q

P

2

i^

i

F

q P

i^

i

P

p

i^ =

i

Q

q

3

i

i F

q

p ∂

= −

3

i

i F

P

Q

4

i

i F

q

p ∂

= −

4

i

i F

Q

P

3

i^

i

F

p Q

i^

i

P

p

=

i^

i

Q

q

=

4

i^

i

F

p P

i^

i

Q

p

i^

i

P

q

=