Transcript Adiabatic Invariance

Adiabatic Invariance
Slow Changes

A periodic system may have
slow changes with time.
• Slow compared to period
• Phase space trajectory
open

What happens to the action?
 (t )   (t   )
  a
a constant
q  q( J , w,  )
p  p ( J , w,  )
S  S (q, w,  )
p
E(t) = H(q,p,t)
q
H  H ( q, p,  ) 
S
t
Change in Action

Find the change in the action
from Hamilton’s equations.
• First two terms sum to zero
• Only the time change of the
principal function remains
H
J  
w
2


H

q

H

p

S 


J  


 q w p w tw 
2


H

q

H

p

S 
J  


 
 q w p w w 
Average Change

Take the time average over one period.
• Assume small changes
• Neglect higher order terms
2
2


S


S
J   
 dw   
dw
  w
  w
1
  S
J   
  

t 
2
S 
2  S

0
  
2
 t 

The action is invariant.
  a
constant
  0
 2  0
Lorentz Force

A moving electron in a
uniform magnetic field has
uniform circular motion.
• Angular frequency wc from
the force.
• A magnetic moment M
relates to the angular
momentum.

The Lagrangian can be
written in terms of M.


dv  qB
v
dt
mc
qB
mc


qJ
M
2mc
wc  
mv2  
L
M J
2
Lagrangian Solution

Write the problem in
cylindrical coordinates.
• z-component is along B.

The angle q is cyclic.
• Constant momentum pq.

Find the radial equation of
motion.
qr 2q
Mz 
2c
m 2 2 2
qBr 2q
2
L  (r  r q  z ) 
2
2c
2
qBr
pq  mr q 
2c
2
z
r
q
qB 

mr  rq mq 
0
c 

Circular Motion

Uniform circular motion limits
variables.
• Radius is constant.
• Angular velocity is constant.
• Magnetic moment is related
to the constants.

q  
qB
 wc
mc
qBr 2
pq  
2c
qr 2wc
q 2 Br 2
Mz 

2c
2mc2
Find the action J.
• Constants times the
magnetic moment
Jq   pq dq
Jq  
qBr 2
c

2mc
Mz
q
Invariance Applied

Adiabatic Invariance applies is the variation of a
variable is slow compared to the period.
• Slow variations in the magnetic field

The magnetic moment is adiabatically invariant.
• B times the area of the orbit is constant
Mz 
q
Jq
2mc
 c
  r 2 B
Jq 
 q 
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