PHYS 4150 Lecture 9: plasma_9Exam
Course CodePHYS 4150
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PHYS4150 — P L A S M A P H Y S I C S
l e c t u r e 9 - m a g n e t i c m i r r o r m a c h i n e s ,a d i a b a t i c
m o m e n t s ,t r a p p i n g
G135, University of Colorado, Boulder
1a d i a b at i c i n va r i a n t s
The presence of adiabatic invariants is actually a common phenomenon, which has
been studied extensively in classical mechanics. Here we follow Landau & Lifschitz
and consider a one-dimensional ﬁnite motion, where
is a parameter describing a
very slow change of the system. Here, slow means slow compared to the period
the cyclic motion, i.e.
. Now, because
is slowly changing, so is the energy
of the system, where
. This implies that the change of energy is a function
of λ, from what follows that there is a combination of Eand λ, a so-called adiabatic
invariant, which remains constant.
be the Hamiltonian of such a system, where again
parameter characterizing the slow change. Then,
Now we average over one cycle
and assume that
does not change on this time
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L E C T U R E 9PHYS4150
and using that ˙q=∂H
By further noting that
We have assumed that
is constant along the integration path, which implies that
E=H(p,q;λ)is constant as well. Differentiating Hwith respect to λgives
After substituting this expression into our expression for the change of the mean
energy we get
This result implies that the adiabatic invariant
remains constant even when the parameter
is changing slowly.
is actually the area
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