
Nonlinear Dynamics
Beam physics is a prime example for the occurance of nonlinear dynamics.
In the large circular accelerators, the individual particles moving near
the speed of light often travel around the ring for billions of turns.
The motion in the tube, which is controlled by magnetic fields, is nonlinear
and exhibits many of the interesting phenomena that are known from dynamical
systems, including chaos.
The picture shows a socalled "Poincare plot" of the motion of a group
of five particle in a repetitive accelerator with horizontal and vertical
degrees of freedom. Every time the particles have completed one orbit around
the system, their positions and momenta are recorded, and this procedure
is repeated for about 1000 turns. In this view, it is very difficult to
estimate the longer term behavior of the particles, in particular regarding
their stability. 


On the other hand, the picture on the right shows a Poincare plot displayed
in socalled normal form coordinates, the result of a complicated iterative
perturbative sequence of nonlinear transformations. In normal form coordinates,
the motion appears beautifully regular, and seems to follows circles. In
these coordinates, it is much easier to estimate the longterm behavior
of the motion. 
However, exactly circular motion in normal form coordinates is tantamount
to the generation of invariants and integrability of the dynamical system.
But practically, integrable systems are very rare; and indeed, magnified
views of the Poincare plot in normal form coordinates reveal very fine
deviations from circularity. The pictures show these deviations,
which here are in the range of parts per million; to the right is a cut
depending on radius and angle of the horizontal coordinates, and below
is a cut depending on the angles of horizontal and vertical motion. 


The detailed analysis and understanding of these phenomena is connected
to cutting edge techniques in the theory of dynamical systems, modern Hamiltonian
mechanics, mathematical physics, and verified computational methods. 
The pictures below show actual measurements of particle motion in an accelerator,
both in cartesian (left) and polar (right) coordinates.
The four figures below represent the evolution of the distribution of
charged particles in a beam as it travels through an accelerator under
the influence of nonlinear forces (z represents relative distance along
the machine). The core of the beam is represented in green. The evolution
of the beam is very much like that of the evolution of a galaxy. The beam
starts as a featureless clump; then spiral arms develop (red), followed
by a circumferential halo (blue). Understanding how to control such complex
dynamics is an important goal of beam physics.
