739 research outputs found
Prediction of long term stability by extrapolation
This paper studies the possibility of using the survival function to predict
long term stability by extrapolation. The survival function is a function of
the initial coordinates and is the number of turns a particle will survive for
a given set of initial coordinates. To determine the difficulties in
extrapolating the survival function, tracking studies were done to compute the
survival function. The survival function was found to have two properties that
may cause difficulties in extrapolating the survival function. One is the
existence of rapid oscillations, and the second is the existence of plateaus.
It was found that it appears possible to extrapolate the survival function to
estimate long term stability by taking the two difficulties into account. A
model is proposed which pictures the survival function to be a series of
plateaus with rapid oscillations superimposed on the plateaus. The tracking
studies give results for the widths of these plateaus and for the seperation
between adjacent plateaus which can be used to extrapolate and estimate the
location of plateaus that indicate survival for longer times than can be found
by tracking.Comment: 23 pages, 15 figure
Normal Mode Tunes for Linear Coupled Motion in Six Dimensional Phase Space
The motion of a particle in 6-dimensional phase space in the presence of
linear coupling can be written as the sum of 3 normal modes. A cubic equation
is found for the tune of the normal modes, which allows the normal mode tunes
to be computed from the 6x6 one turn transfer matrix. This result is similar to
the quadratic equation found for the normal mode tunes for the motion of a
particle in 4-dimensional phase space. These results are useful in tracking
programs where the one turn transfer matrix can be computed by multiplying the
transfer matrices of each element of the lattice. The tunes of the 3 normal
modes, for motion in 6-dimensional phase space, can then be found by solving
the cubic equation. Explicit solutions of the cubic equation for the tune are
given in terms of the elements of the 6x6 one turn transfer matrix.Comment: 3 pages, gzipped postscript paper (77k
Linear Orbit Parameters for the Exact Equations of Motion
This paper defines the beta function and other linear orbit parameters using
the exact equations of motion. The orbit functions are redefined using the
exact equations. Expressions are found for the transfer matrix and the
emittances. Differential equations are found for the beta function and the eta
function. New relationships between the linear orbit parameters are found.Comment: 14 pages, gzipped postscript paper (120k
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