Transcript Poisson Brackets - Northern Illinois University

Poisson Brackets
Matrix Form

The dynamic variables can
be assigned to a single set.
• q1, q2, …, qn, p1, p2, …, pn
• z1, z2, …, z2n

Hamilton’s equations can be
written in terms of za
 0
J  
 I
I

0
za  J a
H ( z, t )
z 
• Symplectic 2n x 2n matrix
• Return the Lagrangian
1 a
z J a z   H ( z , t )
2
d 1

 p j q j  H ( z , t )   p j q j 
dt  2

L( z, z, t ) 
Dynamical Variable
dF F ( z, t )
F ( z, t )

za 
dt
za
t
dF F
H F

Ja

dt za
z t



A dynamical variable F can
be expanded in terms of the
independent variables.
S
This can be expressed
in
terms of the Hamiltonian.
1
The Hamiltonian provides
knowledge of F in phase
space.
Angular Momentum
Example
 The two dimensional
harmonic oscillator can be
put in normalized
coordinates.
• m=k=1

Find the change in angular
momentum l.
• It’s conserved

 
H  12 p12  p22  12 q12  q22

l  q1 p2  q2 p1
dl l
H

J a
dt za
z 
dl l H l H


dt q i pi pi q i
dl
 p2 p1  p1 p2  q2 q1  q1q2  0
dt
Poisson Bracket
F , H   F
za
J a
H
z 
A, B  A Ja B
za
z
A, B 
za 
A B B A

i
q pi pi q i
za
H
J a
 za , H 
za
z 



The time-independent part of
the expansion is the Poisson
bracket of F with H.
S
This can be generalized
for
any two dynamical variables.
1
Hamilton’s equations are the
Poisson bracket of the
coordinates with the
Hamitonian.
Bracket Properties

The Poisson bracket defines
the Lie algebra for the
coordinates q, p.
• Bilinear
{A + B, C} ={A, C} 1+ {B, C}
S
{kA, B} = k{A, B}
• Antisymmetric
{A, B} = {B, A}
• Jacobi identity
{A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0
Poisson Properties

In addition to the Lie algebra
properties there are two
other properties.
• Product rule
• Chain rule

The Poisson bracket acts
like a derivative.
za , z  Ja
A, BC  {A, B}C  B{A, C}
A( , t ), B( , t ) 
A
B
J 
z
z
  B
A a
A, B 
J 
a z
z  
A, B 
A
a ,    B
a
 
Poisson Bracket Theorem
d
A, B  A, B, H   A, B
dt
t
A, B, H   A, H , B A, B, H 



A, B    A Ja B 
t
t  za
z  
2 A
B A
2B

J a

J a
za t
z  za
z  t
 A   B 
  , B    A, 
 t   t 

Let za(t) describe the time
development of some
system. This is generated by
a Hamiltonian if and only if
every pair of dynamical
variables satisfies the
following relation:
 
d
A, B  A , B  A, B 
dt
Not Hamiltonian


Equations of motion must
follow standard form if they
come from a Hamiltonian.
Consider a pair of equations
in 1-dimension.
q  pq
p   pq
CeCt
q  q0
p0  q0eCt
C
p  p0
p0  q0 eCt
C  p0  q0  p  q
H
q 
 pq
p
2H
p
pq
H
 p 
 pq
q
2H
q
qp
Not consistent with H
q, p  0
d
q, p  q, p q, p 
dt
 qp, p q, qp  q, pp  qq, p
 p  q Not consistent with motion
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