Transcript Slide 1
Hamiltonian Formalism
g f dg
(
x
, •
f y
)
ux vdy
df
Legendre transformations
dg xdu
f
x df
g
y dx
dy
f
y
udx
g
u dy xdu du
Legendre transformation :
udx
vdy u v
f
x
f
y udx g
(
y
,
vdy
udx u
)
v xdu
g y x
g
u g
(
y
,
u
)
v
f
(
x
,
y
)
ux
g
y
;
x
g
u
Adrien-Marie Legendre (1752 –1833)
H
What is H?
H
M m
1
L
q
m q
m
L
•
Conjugate momentum
L
q
m
•
Then
L
(
q
1 ,...,
q M
,
q
1 ,...,
q
M
,
t
)
m M
1 •
So
p m v g
(
y
,
u
)
f
(
x
,
y
)
ux
g
y
;
x
g
u p m q
m H
(
q
1 ,...,
q M
,
p
1 ,...,
p M
,
t
)
L
(
q
1 ,...,
q M
,
q
1 ,...,
q
M
,
t
)
m M
1
p m q
m
dH
What is H?
H
L
m M
1
p dL
d
L
t m M
dt
1
p m m M
1
q
m
p m
d q
M m
m
1
m
q
q
m m
L
q m dp m
dq m
L
q
m
L
q
m
d q
m
p m
q m
L d dt
d dt
L
q
m
m
m
m M
1
m dq m
q
m dp m
L
t dt
What is H?
dH
m M
1
m dq m
q
m dp m
L
t dt
H
q m
m
H
p m
q
m
H
t
L
t
dH dt dH dt
M m
1
H
q m q
m
H
p m
m
H
t
m M
1
m q
m
q
m m
H
t dH dt
H
t
What is H?
•
If
L
L
0 (
q
1 ,
q
2 ,...,
q M
,
t
)
i l
1
i
(
q
1 ,
q
2 ,...,
q M
i
,
j l
2
ij
(
q
1 ,
q
2 ,...,
q M
,
t
)
q
i q
j
L
0
L
1
L
2 ,
t
)
q
i
•
Then
H
L
2
L
0 • •
Kinetic energy
T
i
In generalized coordinates
1 2
m i
r i
r i
2
r i
(
q
1 ,
q
2 ,...,
q M
,
t
)
T
i
1 2
m i
r
i
t
j
r
i
q j q
j
2
T
i
j i
1 2
m i m i
r
i
q
r
t
i j T
r
i
j
What is H?
r
i
q j q
j
2
i
t q
j
1 2
m i j
,
k i
1 2
m
r i
i
q j
• • •
For scleronomous generalized coordinates
1 2
j
,
k i m i r
i
r
i
q j r i
(
q
, 1
r
i
q k q
2 ,...,
q
j q
k q M
)
L
2
Then If
V
H
L
0
L
2
L
0
T
L
0
H
T
V
r
i
t
r
i
q k
2
q
j q
k E mec
What is H?
L
L
0
L
1
L
2 •
For scleronomous generalized coordinates, H is a total mechanical energy of the system (even if H depends explicitly on time)
•
If H does not depend explicitly on time, it is a constant of motion (even if is not a total mechanical energy)
•
In all other cases, H is neither a total mechanical energy, nor a constant of motion
•
Hamiltonian : Hamilton’s equations
H
(
q
1 ,...,
q M
,
p
1 ,...,
p M
,
t
)
L
(
q
1 ,...,
q M
,
q
1 ,...,
q
M
,
t
)
M m
1
p m q
m
•
Hamilton’s equations of motion :
q
m
H
p m m
H
q m
H
t
L
t
Sir William Rowan Hamilton (1805 – 1865)
Hamiltonian formalism
•
For a system with
2M M
degrees of freedom, we have independent variables
q
and
p
:
2M
-dimensional phase space (vs. configuration space in Lagrangian formalism)
•
Instead of
M
second-order differential equations in the Lagrangian formalism we work with
2M
first-order differential equations in the Hamiltonian formalism
•
Hamiltonian approach works best for closed holonomic systems
•
Hamiltonian approach is particularly useful in quantum mechanics , statistical physics , nonlinear physics , perturbation theory
d dt
L
q
j
Hamiltonian formalism for open systems
L
q j
Q j dp j dt
L
q j
Q j q
m
H
p m m
H
q m
Q m
Hamilton’s equations in symplectic notation
•
Construct a column matrix (vector) with
2M
elements
j
q j
;
M
j
p j
•
Then
H
η
j
H
q j
;
H
η
M
j
H
p j
J
•
0 Construct a
2M
x
2M
1 1 0
1
1 0 ...
0 0 1 ...
0
square matrix as follows:
...
...
...
...
0 0 ...
1
0
0 0 ...
0 0 0 ...
0 ...
...
...
...
0 0 ...
0
Hamilton’s equations in symplectic notation
•
Then the equations of motion will look compact in the symplectic ( matrix ) notation :
J
H
η
•
Example (
M
= 2):
q
q
2 1 1 2 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0
H
H H
H
/ / / /
q q p
1
p
2 1 2
Lagrangian to Hamiltonian
• •
Obtain conjugate momenta from a Lagrangian
p m
L
m
Write a Hamiltonian
•
q
q
H
L
m M
1
p m
m
(
q
,...,
q
,
p
,...,
p
,
t
) •
q
q
(
q
,...,
q
,
p
,...,
p
,
t
)
Hamiltonian to make it a function of coordinates, momenta, and time
Lagrangian to Hamiltonian
b mn
b nm
•
L
(
For a Lagrangian quadratic in generalized velocities
q
1 ,...,
q M
,
q
1 ,...,
q
M
, )
L
0 (
q
1 ,...,
M
,
t
) • •
m a m
(
q
1 ,...,
q M
,
t
Then a Hamiltonian
)
q
m
Write a symplectic notation:
H
m
n
,
p
b mn
L L
(
q
1 ,...,
L
0
q M
~
a
,
t
)
q
1 2
m q
~
q
n
b
q p
L
0
q a
1 2 ~
q b
~ (
p
a
)
L
0 1 2 ~
q b
p
•
Conjugate momenta
L
~
q
q
L
0 ~
a
1 2 ~
q b
a
b
Lagrangian to Hamiltonian p
a
b
•
Inverting this equation p
a
b
b
1 (
p
a
) ( )
b
1 •
Then a Hamiltonian
( )
b
1 (
p
a
)
L
0
H
~ (
p
a
)
L
0 1 2 ~
q b
1 2 ( )
b
1
bb
1 (
p
a
)
H
1 2 ( )
b
1 (
p
a
)
L
0
Example: electromagnetism
L
m
(
r
r
)
q
2
q
(
r
A
)
p j
L
r
j
m r
j
qA j r
j
p j H
p
r
L
qA j m r
p
q A m H
p
p
q A
m m
2
p m
q A
2
q
q
p
q A
m
A
Example: electromagnetism
H q
p
p
p
p
A m
p
q p
m
q A
q A
A
m
p
2
A
2
p
p
m
p
p
q
2
A
p
2
p m
2
m
q
2 2
A m
A
2
m
q
p
m q
2
q p
A
A
2 2
q
q p
2
m
m
A
2
A
q q
2 2
q p H
A
A q
A
p q
m
q A
2
p
2
q m
2 2
m
A
q A
2
A
A
q
Hamilton’s equations from the variational principle
•
Action functional :
t
2
I
t
1
L
(
q
1 ,...,
q M
,
q
1 ,...,
q
M
,
t
)
dt
t t
1 2
m M
1
p m q
m
H
(
q
1 ,...,
q M
,
p
1 ,...,
p M
,
t
)
dt
•
Variations in the phase space :
q m
(
t
, )
p m
(
t
, )
q m
0
p m
0 (
t
) (
t
)
m
m
(
t
) (
t
)
m m
(
t
1 ) (
t
1 )
m m
(
t
2 (
t
2 ) ) 0 0
Hamilton’s equations from the variational principle
dI d
t t
1 2
M t t
1 2
m M
1
m
1
m dp d
m q
m
•
q
m
m
Integrating by parts
p m d q
d
m
p m
H
q m
m dH d
dt
H
p m
m
dt t
1 2
t M
m
1
m q
m t
1 2
t m M
1
m
m
m
H
p m
H
q m
m
m
H
p m
m
m
H
q m
dt
m m M
1
m
dt p m t
2
t
1
Hamilton’s equations from the variational principle
dI d
t t
1
m M
2 1
q
m
H
p m
m
m
H
q m
m dt
0 •
For arbitrary independent variations
q
m
H
p m
;
m
H
q m
Conservation laws
m
H
q m
;
m
H
p m
;
dH dt
H
t
•
If a Hamiltonian does not depend on a certain coordinate explicitly (cyclic), the corresponding conjugate momentum is a constant of motion
•
If a Hamiltonian does not depend on a certain conjugate momentum explicitly (cyclic), the corresponding coordinate is a constant of motion
•
If a Hamiltonian does not depend on time explicitly, this Hamiltonian is a constant of motion
•
Higher-derivative Lagrangians
Let us recall: L
dxdydz
L
L
d i i η m
(
x n i r m
(
t
)
x n
/ ) ,
dt i x n
,
t
i x n r m
0 , 1 , 2 , 3 ...
?
x
,
y
,
z
,
t
,...
?
r x
,
r y
,
r z
•
Lagrangians with i > 1 occur in many systems and theories: 1. Non-relativistic classical radiating charged particle 2.
(see Jackson) Dirac’s relativistic generalization of that 3. Nonlinear dynamics 4. Cosmology 5. String theory 6. Etc.
Higher-derivative Lagrangians
dI d
•
For simplicity, consider a 1D case:
•
Variation
L
L
(
x
,
x
(
t
, )
x
,
x
,
x
,...)
x
0 (
t
) (
t
)
t t
1 2
dL dt d
t t
1 2
L
x
L
x
L
x
L
x
...
dt t
1
t
2
L
x
dt
t
1
t
2
d dt
L
x
dt
L
x
t
2
t
1 Mikhail Vasilievich Ostrogradsky (1801 - 1862)
dI d
t t
1 2
Higher-derivative Lagrangians
dL dt d
t t
1 2
L
x
L
x
L
x
L
x
...
dt t
1
t
2
L
x
dt
t
1
t
2
d dt
L
x
dt
L
x
t
2
t
1
t
1
t
2
d
2
dt
2
L
x
dt
L
x
t
2
t
1
d dt
L
x
t
2
t
1
dI d
t t
1 2
Higher-derivative Lagrangians
dL dt d
t t
1 2
L
x
L
x
L
x
L
x
...
dt t
1
t
2
L
x
dt
t
1
t
2
d dt
L
x
dt
L
x
t
2
t
1
t
1
t
2
d
2
dt
2
L
x
dt
L
x
t
2
t
1
d dt
L
x
t
2
t
1
t
1
t
2
d
3
dt
3
L
x
dt
L
x
t
2
t
1
d dt
L
x
t
2
d
2
dt
2
t
1
L
x
t
2
t
1
dI d
t t
1 2
L
x
Higher-derivative Lagrangians
d dt
L
x
d dt
2 2
L
x
d dt
3 3
L
x
...
dt
L
x
d dt
L
x
d
2
dt
2
L
x
...
t
2
t
1
L
x
d dt
L
x
...
L
x
...
t
2
t
1 ...
0
t
2
t
1
p i
k
...
i q
1 •
Generalized coordinates/momenta:
x
;
q
2
x
;
q
3
x
;
q
4
x
; ...
d dt p
1
p
2
p
3 ...
k
i
L dt k x
dI d
t t
1 2
k
1
Higher-derivative Lagrangians
d dt k
1
L
q k
dt
k
1
p k d dt k
1
t
2
t
1 •
Euler-Lagrange equations:
0
k
1
d dt k
1
L
q k
0 •
We have formulated a ‘higher-order’ Lagrangian formalism
•
What kind of behavior does it produce?
q
1
x q
2
x
p
1
p
2
L
x
L
x
d L
Example
L
(
x
,
x
,
x
)
L dt
x
p
1
L
x
p
1 2
L
x
2
L
x
d dt
L
x
d dt
2
L
x
0
L
x
1
H
p
1 1
p
2 2
L
(
x
, ,
x
)
dH
H
p
1 1
p
1
d q
1
q
1
dp
1
Example
p
2 2
L
(
x
, ,
x
)
p
2
d q
2
q
2
dp
2
L
x dx
L
x
d
x
p
1
dq
2
p
1 2
q
1
dp
1
dq
2
p
2
p
2
d q
2
d q
2
q
2
dp
2
L
x
L
x
d x
1
dq
1 1
dp
1 2
dp
2 1
dq
1 2
dq
2 1
L
x
p
2
q
m
H
p m
;
m
H
q m
;
dH dt
0
L
x
p
1 2
H
Example
p
1
q
1
p
2
q
2
L q
1
x
q
2 •
H is conserved and it generates evolution – it is a Hamiltonian!
H
p
1
q
1
p
2
q
2
L
p
1
q
2
p
2 2
L
•
Hamiltonian linear in momentum?!?!?!
•
No low boundary ground state!!!
on the total energy – lack of
•
Produces ‘runaway’ solutions: the system becomes highly unstable - collapse and explosion at the same time
‘Runaway’ solutions
•
Unrestricted low boundary of the total energy produces instabilities
•
Additionally, we generate new degrees of freedom, which require introduction of additional (originally unknown) initial conditions for them
•
These problems are solved by means of introduction of constraints
•
Constraints restrict unstable behavior and eliminate unnecessary new degrees of freedom
Canonical transformations
•
Recall gauge invariance (leaves the evolution of the
•
system unchanged):
L
'
L
Let’s combine
dF L
{
q m
};
L
'
dt
gauge invariance with Legendre
{
Q m
}
m M
1
transformation:
M p m q
m
H
m
1
P m
m
K
dF dt
m
•
K P m
;
m
K
Q m
;
dK dt
K t
K – is the new Hamiltonian (‘Kamiltonian’
)
•
K may be functionally different from H
9.1
m M
1
Canonical transformations
p m q
m
H
m M
1
P m
m
K
dF dt
•
Multiplying by the time differential:
m M
1
dF p m dq m
Hdt
m M
1
p m dq m
m M
1
P m dQ m
P m dQ m
•
So
p m
F
q m
;
P m
F
Q m
;
K
Kdt
dF K
H
dt
H
F
t F
F
(
q
1 ,
q
2 ,...,
q M
,
Q
1 ,
Q
2 ,...,
Q M
,
t
) 9.1
Generating functions
F
F
1 (
q
1 ,
q
2 ,...,
q M
,
Q
1 ,
Q
2 ,...,
Q M
,
t
) •
Such functions are called generating functions canonical transformations of
•
They are functions of both the old and the new canonical variables, so establish a link between the two sets
•
Legendre transformations may yield a variety of other generating functions
9.1
9.1
Generating functions
F
F
1 (
q
1 ,
q
2 ,...,
q M
,
Q
1 ,
Q
2 ,...,
Q M
,
t
) •
F
We have three additional choices:
F
2 (
q
1 ,
q
2 ,...,
q M
,
P
1 ,
P
2 ,...,
P M
,
t
)
m M
1
Q m P m F
F
F
3 (
p
1 ,
F
4 (
p
1 ,
p
2 ,...,
p
2 ,...,
p M p M
,
Q
1 ,
Q
2 ,...,
Q M
,
P
1 ,
P
2 ,...,
P M
,
t
) ,
t
)
m M
1
q m m M
1
q m p m p m
Q m P m
•
Canonical transformations may also be produced by a mixture of the four generating functions
F
1
m M
1
q m Q m
An example of a canonical transformation
p m
F
q m
;
P m
F
Q m
;
K
H
F
t
9.2
Q m
p m
;
P m
q m
;
K
H
•
Generalized coordinates are indistinguishable from their conjugate momenta, and the nomenclature for them is arbitrary
•
Bottom-line: generalized coordinates and their conjugate momenta should be treated equally in the phase space
Criterion for canonical transformations
Q
Q
(
q
,
p
);
P
P
(
q
,
p
)
q
q
(
Q
,
P
);
p
p
(
Q
,
P
) 9.4
•
How to make sure this transformation is canonical?
q
Q q
Q p
H p
Q q
H q
Q p
•
On the other hand
H
P
H
p
p
P
H
q
q
P
Q
•
q
If
p
P
•
Then
;
Q
p
q
P
H
P
Criterion for canonical transformations
9.4
•
Similarly,
P
q q
P
p
H
p
P
q
H
q
P
p
H
Q
H
p
p
Q
H
q
q
Q
•
If
•
Then
P
q
p
Q
;
P
p
q
Q
H
Q
Criterion for canonical transformations
9.4
•
So,
q
H
p
;
H
q
H
P
;
H
Q
; •
If
P
p
q
Q
;
Q
q
p
P
;
P
q
p
Q
;
Q
p
q
P
9.4
i
Canonical transformations in a symplectic form
J
H
η
•
After transformation
j
q j
;
M
j
ζ
ζ
(
η
)
p j j
Q j
;
M
j
2
j M
1
i j
j A ij
i j
A
AJ
H
η
P j
•
On the other hand
H
i
2
j M
1
H
j
i j
H
η
~
A
H
ζ
F
t
ζ
AJ
~
A
H
ζ
0
K
H
Canonical transformations in a symplectic form ζ
AJ
~
A
H
ζ
•
For the transformations to be canonical: ζ
J
H
ζ
•
Hence, the canonicity criterion is:
~
AJ A
J
9.4
•
For the case M = 1, it is reduced to (check yourself)
P
p
q
Q
;
Q
q
p
P
;
P
q
p
Q
;
Q
p
q
P
1D harmonic oscillator
P H
•
H
p
2
kq
2
E
2
m
2 •
Let us find a conserved canonical momentum
const
0
H Q H
H
(
P
)
Generating function
F
F
1 (
q
,
Q
)
P
F
Q p
F
q
p
2 2
m
kq
2 2
H
F
Q
F
/ 2
m
q
2
kq
2 2 9.3
9.3
•
H
1D harmonic oscillator
F
/
Q
F
/
q
2
m
2
kq
2 2
Nonlinear partial differential equation for F
H
dA dQ
• •
A
'
Let’s try to separate variables
F
A
(
Q
)
b
(
q
)
H
Let’s try
2
q
2
b
'
A
2 2
m
q k
2
q
2
H
(
P
)
P A
'
b
2
m
' 2
A
'
A
2
m
k
A
2
km m Q
m
A
2
dA
km
k
cot 1
A
2
kq
2 2
km m A km
Q
F m
/
k
cot
1D harmonic oscillator
1
A
/
km
A
km
cot
Q
A
(
Q
)
b
(
q
)
km
cot
Q P H
• (
We found a generating function!
P
)
P E
H
P
1
F
Q
2 sin 2 (
q Q
2
k k
/
m
)
k
/
m
q
2 2
k
/
m
Q
t
t
0 9.3
q
2
E
sin((
t
t
0 )
k k
/
m
)
F
p km
cot
F
q
Q
1D harmonic oscillator
k
/
q m
q
2 2
km q
cot
Q k
2
k
/
m
sin((
t
t
0 )
Q
2
Em
cos((
t
t
0 )
k
/
m
)
t k
/
m
)
t
0 9.3
q
p
2
E
sin((
t
t
0 )
k k
2
Em
cos((
t
t
0 ) /
m
)
k
/
m
)
p
2 / 2
m
kq
2 / 2
E
q
,
p
1D harmonic oscillator
9.3
Q
,
P
Canonical invariants
• •
What remains invariant after a canonical transformation?
d
i
Matrix A is a Jacobian
2
j M
1
A ij d
j
ζ
A ij
ζ
(
η
)
i
j
of a space transformation
9.5
•
From calculus, for elementary volumes: A
2
i M
AJ J
1
d
i
det(
A ij
) 2
i M
1
d
i
A
• ~
A Transformation is canonical if
~
A
J J
0 1 1 0
J
2
i M
d
1 ~
AJ A
1
A
i
~
A J
1
A
A
1
A
2 ~
A
1
Canonical invariants
2
i M
1
d
i
A
2
i M
1
d
i
2
i M
1
d
i
•
For a volume in the phase space
V
2
M
i
1
d
i
2
M
i
1
d
i
V
•
Magnitude of volume in the phase space is invariant with respect to canonical transformations:
V
V
9.5
9.5
Canonical invariants
v u
•
What else remains invariant after canonical
transformations?
v
(
η
,
t
)
v
η
~
A
u
(
η
,
t
)
u
η
v
ζ
~
A
u
ζ
A
~
u
η
ij
~
A
i j
~
u
ζ
ζ
~
AJ A
~
u
ζ
ζ
(
η
)
J
A
~
u
η
J
v
η
~
u
ζ
A J
~
A
v
ζ
~
u
ζ
J
v
ζ
u
q
u
η
J
v
η
~
u
ζ
J
v
ζ
• •
For M = 1
u
p
0 1 1 0
v
q v
p
For many variables
~
u
η
J
v
η
i
u
q i
u
q
v
p i
v
q i
u
p
v
p
v q
u
q
v
q
v
p
u
p
u
p i
9.5
9.5
Poisson brackets
•
Poisson brackets :
~
u
η
J
v
η
i
u
q i
v
p i
v
q i
u
p i
[
u
,
v
] •
Poisson brackets are invariant with respect to any canonical transformation
i
u
q i
p v i
v
q i
p u i
i
u
Q i
v
P i
v
Q i
u
P i
Siméon Denis Poisson (1781 – 1840)
Poisson brackets
•
Properties of Poisson brackets :
[
F
,
F
] 0 [
F
,
G
] [
G
,
F
] [
F
,
G
X
] [
F
,
G
] [
F
,
X
] [
F
,
GX
] [
F
,
G
]
X
G
[
F
,
X
] [
aF
bG
,
X
]
a
[
F
,
X
]
b
[
G
,
X
] [
F
, [
G
,
X
]] [
G
, [
X
,
F
]] [
X
, [
F
,
G
]] 0 9.5
[
η
,
η
]
Poisson brackets
~
η
η
J
η
η
1 J 1
J
•
In matrix element notation:
[
q i ,q j
] 0 [
q i ,p j
]
ij
[
η
,
η
]
lm
[
p i ,p j
] 0 [
η l ,η m
] [
p i ,q j
]
J lm
ij
•
In quantum mechanics , for the commutators of coordinate and momentum operators:
9.5
[
q
ˆ
i , q
ˆ
j
] [
i , p j
] 0 [
q
ˆ
i ,
ˆ
j
] [ ˆ
i , q
ˆ
j
]
i
ij
u
u
(
η
,
t
)
Poisson brackets and equations of
du
J
dt
H
η
~
u
η
u
t du dt
motion
~
u
η
J
H
η
u
t
u
t
[
u
,
H
] [
u
,
H
]
u
t
9.6
dH dt d
η
dt
H
t
η
t
[
H
,
H
[
η
,
H
] ]
dH dt
H
t
[
η
,
H
]
Poisson brackets and conservation
du dt
laws
u t
•
If
u
is a constant of motion
[
u
,
H
]
du
/
dt
0
u
t
[
H
,
u
] •
If
u
has no explicit time dependence
[
H
,
u
] 0 •
In quantum mechanics , conserved quantities commute with the Hamiltonian
9.6
Poisson brackets and conservation laws
9.6
•
If
u
and
v
are constants of motion with no explicit time dependence
[
H
,
u
] 0 ; [
H
,
v
] 0 •
For Poisson brackets:
[
F
, [
G
,
X
]] [
G
, [
X
,
F
]] [
X
, [
F
,
G
]] 0 [
u
, [
v
,
H
]] [
v
, [
H
,
u
]] [
H
, [
u
,
v
]] 0
d
[
u
,
v
] 0
dt
•
If we know at least two constants of motion, we can obtain further constants of motion
Infinitesimal canonical transformations
9.4
F
•
Let us consider a canonical transformation with the following generating function (
ε
– small parameter):
G
(
q
1 ,
q
2 ,...,
q M
,
P
1 ,
P
2 ,...,
P M
)
m M
1 (
q m
Q m
)
P m m M
• 1
Then
p m q
m
H
m M
1
P m
m
K
dF dt
m M
1
P m
m
K
m M
1
G
q m q
m
G
P m
m
m M
1 (
q
m
m
)
P m
m M
1 (
q m
Q m
)
m
Infinitesimal canonical transformations
9.4
m M
1 •
Multiplying by dt
p m dq m
Hdt
m M
1
P m dQ m
K dt
m M
1
G
q m dq m
G
P m dP m
m M
1 (
P m dq m
P m dQ m
)
m M
1 (
q m dP m
Q m dP m
)
P m
•
Then
p m
G
q m Q m
q m
G
P m K
H
Infinitesimal canonical transformations
9.4
P m
ζ
[
u
, •
Infinitesimal canonical transformations :
•
η
p m
In symplectic notation:
J
G
q m
G
η
;
Q m
q m v
] ~
u
η
J
v
η ζ
η
G
P m
A J
ζ
η
G
η
q m
1 ζ
[
η
,
v
]
1 J
v
η
[
η
,
G
]
J
G
η
G
p m
η
η
J
[
η
, 2 ...
G
η
G
]
ζ
η
[
η
,
G
]
dt G
H
Evolution generation ζ
η
[ ,
H
]
dt
η
[
η
,
H
]
d
η ζ
η
d
η
9.6
•
Motion of the system in time interval dt can be described as an infinitesimal transformation generated by the Hamiltonian
•
The system motion in a finite time interval is a succession of infinitesimal transformations, equivalent to a single finite canonical transformation
•
Evolution of the system is a canonical transformation!!!
Application to statistical mechanics
•
In statistical mechanics we deal with huge of particles numbers
•
Instead of describing each particle separately, we describe a given state of the system
•
Each state of the system represents a point phase space in the
•
We cannot determine the initial conditions exactly
•
Instead, we study a certain phase volume – ensemble – as it evolves in time
9.9
Application to statistical mechanics
•
Ensemble can be described by its density – a number of representative points in a given phase volume
D
N V
•
The number of representative points does not change
N
const
•
Ensemble evolution can be thought as a canonical transformation generated by the Hamiltonian
•
Volume of a phase space is a constant canonical transformation for a
V
const
9.9
Application to statistical mechanics
• •
Ensemble is evolving so its density is evolving too
dD dt
D
t
[
D
,
H
]
const
On the other hand
D
N
const V const
D
t
[
H
,
D
] •
Liouville’s theorem
•
In statistical equilibrium
D
t
0 [
H
,
D
] 0 Joseph Liouville (1809 -1882) 9.9
Hamilton –Jacobi theory
•
We can look for the following canonical transformation, relating the constant (e.g. initial) values of the variables with the current ones:
q i
0
p i
0
q i
0 (
q
1 ,...,
q M
,
p i
0 (
q
1 ,...,
q M
,
p
1 ,...,
p
1 ,...,
p M
,
t
)
p M
,
t
) •
The reverse transformations will give us a complete solution
q p i i
q i
(
q
10 ,...,
q M
0 ,
p i
(
q
10 ,...,
q M
0 ,
p
10 ,...,
p M
0 ,
t
)
p
10 ,...,
p M
0 ,
t
) 10.1
10.1
Hamilton –Jacobi theory
p i
• • • •
Let us assume that the Kamiltonian is identically zero Then
K
0
i Q i
K
/
const
P i
;
P i
0 ;
i const
;
K
/
Q i
Choosing the following generating function
F
F
2 (
q
1 ,...,
q M
,
P
1 ,...,
P M
,
t
)
Q i i
Then, for such canonical transformation:
P i
0 ;
F
2 /
q i Q i
F
2 /
P i K
H H H
(
q
1 ,..., (
q
1 ,...,
q M q M
, ,
p
1 ,...,
F
2
q
1
p
,...,
M
,
t
)
F
2
q M
,
t
)
F
2 /
t
F
2
t
0 0
F
2 /
t
10.1
H
(
q
1 ,...,
q M
,
Hamilton –Jacobi theory
F
2
q
1 ,...,
F
2
q M
,
t
)
F
2
t
0
H
• •
Hamilton –Jacobi equation
S
Sir William Rowan Hamilton
F
2 (1805 – 1865)
Hamilton’s principal function
S
S
S
(
q
1 ,...,
q M
,
q
1 ,...,
q M
,
t
)
t
0 •
Partial differential equation
•
First order differential equation
•
Number of variables: M + 1
Karl Gustav Jacob Jacobi (1804 – 1851)
Hamilton –Jacobi theory
H
(
q
1 ,...,
q M
,
S
/
q
1 ,...,
S
/
q M
,
t
)
S
/
t
0 •
Suppose the solution exists, so it will produce M + 1 constants of integration:
S
S
(
q
1 ,...,
q M
, 1 ,...,
M
1 ,
t
) 10.1
H
•
One constant is evident:
S
(
q
1 ,...,
S
/
t
0
S
(
q
1
q
,...,
M
,
q M
1 , ,..., 1 ,...,
M
1
M
, ,
t t
) )
M
1 •
We chose those M constants to be the new momenta
P i
i
•
While the old momenta
p i
S
(
q
1 ,...,
q M
,
q i
1 ,...,
M
,
t
)
Hamilton –Jacobi theory
•
We relate the constants with the initial values of our old variables:
p i
0
S
(
q
1 ,...,
q M
,
q i
1 ,...,
M
,
t
)
q i
q i
0 ;
t
t
0 •
The new coordinates are defined as:
Q i
i
S
(
q
1 ,...,
q
,
M
1
i
,...,
M
,
t
)
q i
q i
0 ;
t
t
0 •
Inverting those formulas we solve our problem
q i p i
q i p i
( 1 ,...,
M
( 1 ,...,
M
, , 1 ,..., 1 ,...,
M
M
,
t
) ,
t
) 10.1
10.1
Have we met before?
dS dt
i
S
q i q
i
S
t p i
S
q i H
S
/
t
0
dS dt
i p i q
i
H
L
•
Remember action?
S
Ldt
const I
t
1
t
2
Ldt
S
(
t
2 )
S
(
t
1 )
10.1
Hamilton’s characteristic function
•
When the Hamiltonian does not depend on time explicitly
dH
/
dt
H
/
t
0 •
Generating function ( Hamilton’s characteristic
Q i p i dW dt
function )
W
P i
W
q i
F
2
W
i
i
W
q i q
i
W
(
q
1 ,...,
q M
, 1 ,...,
M
)
i H p i q
i
H
(
q
1 ,...,
q M
,
W
q
1
W
i p i dq i
,...,
W
q M
const
)
•
Now we require:
10.3
Hamilton’s characteristic function
H
(
q
1 ,...,
q M
,
W
q
1 ,...,
W
q M
) 1 •
So:
K
H
W
t K
1
i
K
Q i
;
i
K
P i
i
0
P i
i
;
i
i
1 ;
Q
1
t
1
W
/ 1 ;
Q i
i
W
/
i
;
i
1 •
Detailed comparison of Hamilton’s characteristic vs. Hamilton’s principal is given in a textbook (10.3)
Hamilton’s characteristic function
H
•
What is the relationship between
S
and
W
?
S
/
t
0
K S H
1
K W
•
One of possible relationships (the most conventional):
10.3
S
(
q
1 ,...,
q M
, 1 ,...,
M
,
t
)
W
(
q
1 ,...,
q M
, 1 ,...,
M
) 1
t H
1 0
Periodic motion
•
For energies small enough we have periodic oscillations ( librations ) – green curves
•
For energies great enough we msy have periodic rotations – red curves
•
Blue curve – separatrix trajectory – transition from librations to rotations bifurcation
10.6
10.6
Action-angle variables
H
(
q
, •
For either type of periodic motion let us introduce a new variable – action variable (don’t confuse with action!):
p
)
p
J p
(
q
,
pdq
)
J
J
( )
W
W
(
q
,
J
) • •
A generalized coordinate conjugate to action variable is the angle variable :
w
W
J
The equation of motion for the angle variable:
H
(
J
)
J
v
(
J
)
const W
S
i
Ldt
const p i dq i
const
Action-angle variables
v w
vt
•
In a compete cycle
w
w
q dq
2
W
q
J dq
J
W
q dq
d dJ
pdq
10.6
d dJ J
1 (
v
(
t
) ) (
vt
) 1
v
v
1 / •
This is a frequency of the periodic motion
w
W
J p
W
q J
pdq
W
10.2
p
H
W
q
Example: 1D Harmonic oscillator
2 1
m
p
2
m H
2 2
q
2 1 2
m
E
;
W
q
2
m
2 2 2
k q
/ 2
m
2
m
1
m
2 2
q
2
S dq
2
m
1
m
2
q
2 2
dq
t
S
m
2
dq
1
m
2 2
q
2
t
1 arcsin
q m
2 2
t
1
Example: 1D Harmonic oscillator
arcsin
q m
2 2
t q
2
m
2 sin(
t
) 10.2
p
W
q
2
m
m
2 2
q
2 2
m
(
p
2
m
2 2
q
2 ) / 2
m
E
cos(
E
t
)
q
2
E m
2 sin(
t
)
p
2
mE
cos(
t
)
W
2
m
1
m
2
q
2 2
dq m
q
0 /
p
0 tan(
t
0 )
J
H
Action-angle variables for 1D harmonic
10.6
pdq p
W
q
2
m
oscillator
2
m
m
2 2
q
2
m
2 2
q
2
dq
2 2 0 cos 2
q
2
m
2 sin
z zdz
2
J
2 •
Therefore, for the frequency:
v
H
J
2
k
2 /
m
Separation of variables in the Hamilton-
10.4
Jacobi equation
•
Sometimes, the principal function can be
•
successfully separated in the following way:
S
i S i
( , ,...,
M
,
t
)
S q i
1
S H
(
q i
,
i
, ,..., ,
t
)
i
0
i
q i
1
M
t
For the Hamiltonian without an explicit time dependence:
S i
W i
i t H i
(
q i
,
W i
q i
, 1 ,...,
M
) •
Functions
H i
may or may not be Hamiltonians
i