Adiabatic Invariance

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Adiabatic Invariance
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Slow Changes

• A periodic system may have slow changes with
  time.
• Slow compared to period
• Phase space trajectory open
• What happens to the action?

a constant
p
E(t) H(q,p,t)
q
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Change in Action

• Find the change in the action from Hamiltons
  equations.
• First two terms sum to zero
• Only the time change of the principal function
  remains

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Average Change

• Take the time average over one period.
• Assume small changes
• Neglect higher order terms
• The action is invariant.

constant
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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.

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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.

z
r
q
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Circular Motion

• Uniform circular motion limits variables.
• Radius is constant.
• Angular velocity is constant.
• Magnetic moment is related to the constants.
• Find the action J.
• Constants times the magnetic moment

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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

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