Single particle behavior in plasmas Page: 22 of 31
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-12-
defined as
[,f] = a + l r .li (4.13)
the result for a general oscillatory system is
-{ ft 1 yk7 1 + 0(c3) 4.14)
As far as plasma theory is concerned we can regard the calculation of adiabatic
invariants as being solved. However, the Hamiltonian formulation is most incon-
venient since H is a function of the potentials (A,0) and not the fields
(B.E). The above method really only required the equation [J.H] = 0 to be
expressed in coordinates which display the phase angle over which we average.
Haas, Hastie, and Taylor have applied the method to the Vlasov equation to -
generate the magnetic moment u, the longitudinal invariant J, and the flux
invarient 0 as given below. The algebra involved was formidable and the
results are worth some comments.
A charged particle in a time independent electromagnetic field will
have as constants of the motion the energy E and the canonical moments
corresponding to any symmetrics of the configuration. If there are no
synetries then, in a strong magnetic field such that rL/L << 1, the mag-
netic moment will be an adiabatic constant:
u - po + ul + 0 ( y(4.15)
where o = v /2B
= + -(9)C + a-(V,-9)b + 4p p1 x b o
and N O x (V 3 + uO), a V 2 .
5 B2
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McNamara, B. Single particle behavior in plasmas, report, March 10, 1977; Livermore, California. (https://gateway.okhistory.org/ark:/67531/metadc1055867/m1/22/?q=%22~1~1~1%22~1: accessed July 16, 2024), The Gateway to Oklahoma History, https://gateway.okhistory.org; crediting UNT Libraries Government Documents Department.