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