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An Introduction to Beam Physics
apparently unstable parts is usually called the dynamic aperture, and it
often looks like a deformed ellipse.
Let us now study a little what conditions seem to favor stable or unstable
motion, respectively. If we divide the phase space regions into parts in which
the nonlinear effects have a tendency to pull particles away from the origin
and those that tend to push the particles toward the origin, then we may
expect that we want to avoid situations where the particles spend too much
time in the “pull away” regions, and it is better if we sample the phase space
uniformly, and thus average out the effects as much as possible.
A nearly uniform sampling of the phase space happens if the linear tune is
not a rational multiple of 2π. On the other hand, if the tune is of the form
μ i = 2πp/q, after q turns the particle will come back to where it was before
and hence can see the same effect, a situation which we call resonance; so
it is at least not a good idea to choose q too small, as repetition after large
numbers of turns is not as critical. The effect of resonances in a circular
accelerator is of great importance to its performance. Chapter 11 is dedicated
to studying this topic in detail.
We may also wonder to what extent it is possible to perform a transformation to normal form coordinates in a similar manner as in the linear case.
As it turns out, most systems cannot be brought to a normal form in which
the motion is exactly circular; the existence of such a transformation is tantamount to the system being integrable, i.e., having one integral of motion per
phase space dimension. Truly integrable systems, however, are very rare. It
turns out, however, there is a powerful order-by-order iterative procedure to
turn a system into nonlinear normal form up to any given order [5]. A simple
example of this procedure is given in Section 11.4.
An Introduction to Beam Physics
apparently unstable parts is usually called the dynamic aperture, and it
often looks like a deformed ellipse.
Let us now study a little what conditions seem to favor stable or unstable
motion, respectively. If we divide the phase space regions into parts in which
the nonlinear effects have a tendency to pull particles away from the origin
and those that tend to push the particles toward the origin, then we may
expect that we want to avoid situations where the particles spend too much
time in the “pull away” regions, and it is better if we sample the phase space
uniformly, and thus average out the effects as much as possible.
A nearly uniform sampling of the phase space happens if the linear tune is
not a rational multiple of 2π. On the other hand, if the tune is of the form
μ i = 2πp/q, after q turns the particle will come back to where it was before
and hence can see the same effect, a situation which we call resonance; so
it is at least not a good idea to choose q too small, as repetition after large
numbers of turns is not as critical. The effect of resonances in a circular
accelerator is of great importance to its performance. Chapter 11 is dedicated
to studying this topic in detail.
We may also wonder to what extent it is possible to perform a transformation to normal form coordinates in a similar manner as in the linear case.
As it turns out, most systems cannot be brought to a normal form in which
the motion is exactly circular; the existence of such a transformation is tantamount to the system being integrable, i.e., having one integral of motion per
phase space dimension. Truly integrable systems, however, are very rare. It
turns out, however, there is a powerful order-by-order iterative procedure to
turn a system into nonlinear normal form up to any given order [5]. A simple
example of this procedure is given in Section 11.4.