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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 tw 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 mr 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 2mc 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 q Jq 2mc c r 2 B Jq q Mz next