Studies of beam dynamics in relativistic klystron two-beam accelerators Page: 65 of 229
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The Poincar6-Cartan form can be expressed as
dw =7) 1-df1+pzdz+ptdt,
where pt = -H is the momentum canonical to time. Provided that the clauses of the
Implicit Function Theorem2 are validated, we can express time as a function of beamline
position, z, and so
dw= d- + pt z- K dz,
dz dz 1.
where the new Hamiltonian, K, is defined as
K(71, 71,t, p,z) = -pz.
The relativistic Hamiltonian in this new parameterization is then given by
K = -qA, - A C - (mc)2 - p - q j). (5.2)
5.2.2 Renormalization Transformation
A second transformation can be performed that render the dynamical variables into a
more convenient form. The relativistic nature of the particle orbits introduces both
momentum (mc) and energy (mc2) scales. The potentials, momenta, and coordinates
are scaled accordingly. The variation of the action is immune to the effects of scalar
multiplication. Hence, we may scale the Poincar6-Cartan form without changing the
resulting equations of motion. We define a new form as
d=d/mc= -1 + Pd(ct)- K
me mc2 mc
or
dJ3 - . dy1 +- ds-K
where K = K/mc, fr = 71/mc, -7 = pt/mc2, and s = ct. Additionally, the poten-
tials are normalized via > = qI/mc2 and A = q /mc. With this normalization the
Hamiltonian function becomes
K = -Az - (-7)2 (5.3)
2In particular, that OH = c is always positive definite.47
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Lidia, Steven M. Studies of beam dynamics in relativistic klystron two-beam accelerators, thesis or dissertation, November 1, 1999; California. (https://digital.library.unt.edu/ark:/67531/metadc724108/m1/65/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.