1. Adiabatic Invariance

2. Slow Changes  A periodic system may have slow changes with time. Slow compared to periodSlow compared to period Phase space trajectory openPhase space trajectory open  What happens to the action? p q E(t) = H(q,p,t) a constant

3. Change in Action  Find the change in the action from Hamilton’s equations. First two terms sum to zeroFirst two terms sum to zero Only the time change of the principal function remainsOnly the time change of the principal function remains

4. Average Change  Take the time average over one period. Assume small changesAssume small changes Neglect higher order termsNeglect higher order terms  The action is invariant. constant

5. Lorentz Force  A moving electron in a uniform magnetic field has uniform circular motion. Angular frequency  c from the force.Angular frequency  c from the force. A magnetic moment M relates to the angular momentum.A magnetic moment M relates to the angular momentum.  The Lagrangian can be written in terms of M.

6. Lagrangian Solution  Write the problem in cylindrical coordinates. z -component is along B.z -component is along B.  The angle  is cyclic. Constant momentum p .Constant momentum p .  Find the radial equation of motion. z r 

7. Circular Motion  Uniform circular motion limits variables. Radius is constant.Radius is constant. Angular velocity is constant.Angular velocity is constant. Magnetic moment is related to the constants.Magnetic moment is related to the constants.  Find the action J. Constants times the magnetic momentConstants times the magnetic moment

8. Invariance Applied  Adiabatic Invariance applies is the variation of a variable is slow compared to the period. Slow variations in the magnetic fieldSlow variations in the magnetic field  The magnetic moment is adiabatically invariant. B times the area of the orbit is constantB times the area of the orbit is constant next