Single Bunch Stability to Monopole Excitation Page: 3 of 20
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Haissinski solution is also used to introduce the action-angle variables that
make the Haissinski Hamiltonian independent of angle, which results in great
simplification of further analysis.
o DO 00
Monopole Dipole Quadrupole
Figure 1: Example contour plots of the lowest three azimuthal modes
The linearized Vlasov equation technique naturally leads to the concept
of azimuthal phase space modes, that are basically the components of the
perturbation to the Haissinski solution with certain azimuthal symmetry.
The first three of such modes are sketched in Fig. 1. Note, that in this figure
and throughout the rest of the paper we assume the simplest phase space
topology, where action-angle variables can be defined uniformly across the
whole plane. In other words, we neglect the possibility of several potential
As seen from Fig. 1 the monopole mode is quite special because, in con-
trast to other modes, its physical space projection does not change signifi-
cantly on the time scale of a synchrotron period. This argues that radiation
rather than Hamiltonian forces define the dynamics of this mode. Also, by
definition of action-angle variables, the unperturbed Haissinski solution has
monopole azimuthal structure. These two features of the monopole mode ex-
plain why it is omitted from the standard linearized Vlasov analysis. Indeed,
in that approach the sole effect of radiation terms in the Fokker-Plank equa-
tion is that they define the Haissinski solution which subsequently cancels
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Podobedov, Boris. Single Bunch Stability to Monopole Excitation, report, January 19, 1999; Menlo Park, California. (digital.library.unt.edu/ark:/67531/metadc613724/m1/3/: accessed November 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.