Maryland Dynamical Systems and Accelerator Theory
The University of Maryland Dynamical Systems and Accelerator
Theory (DSAT) Research Group carries out long-term research in the general
area of Dynamical Systems with a particular emphasis on applications to
Accelerator Physics. Our work,
which is supported in part by the U.S. Department of Energy grant
DEFG0296ER40949, is devoted both to the development of new methods and to the
application of these methods to problems of current interest in accelerator
physics including the theoretical performance of present and proposed high-energy
machines. A major goal of the work
of the DSAT Group is to describe, analyze, predict, and understand the linear
and nonlinear behavior of single- and multi-particle systems. Considerable progress has been made in
the single-particle case. Map
methods employing Lie-algebraic and truncated-power-series-algebraic tools have
been developed to compute and analyze both linear and high-order nonlinear
behavior for machines with idealized beam-line elements. These methods are currently being extended to treat real beam-line elements including
high-order multipole and fringe-field effects. As outlined below, additional work is planned on the
single-particle case as well as generalizations to the interacting multi-particle
case. Finally, where appropriate
and when opportune, research is carried out in other areas of the broad field
of Dynamical Systems. Examples of
such areas include light (both ray and wave) optics, electron microscopes, and
map/operator methods applied to various problems in the rapidly developing
field of Quantum Computing/Information.
addition to its research effort, the DSAT Group is actively engaged in the
education of students and post-doctoral research associates. To this end, it presents a regular
graduate seminar course in accelerator physics, directs graduate students in
M.S. and Ph.D. thesis research, and guides and fosters the research of
Post-doctoral Research Associates and Visiting Scientists.
Past, Present, and Future Research Areas
(Hamiltonian) Orbit Theory
approximation including coupling between all degrees of freedom.
properties of machines with idealized beam-line elements.
for real (non-idealized) beam-line elements based on 3-dimensional field
data and including all nonlinear multipole and fringe-field effects.
and tools for analysis of nonlinear behavior including normal-form
code for computing and analyzing nonlinear behavior of real machines with
of multi-particle distributions and distribution functions in absence of
Construction of moment invariants for the symplectic group.
of spin, including maps for all standard (idealized and real) elements and
prediction of long-term behavior (e.g., dynamic aperture) based on
properties of one-turn map.
construction of generalized moment invariants for nonlinear symplectic
and application of symplectic geometry: Symplectic tracking. Equivalence classes of homogeneous polynomials under
the action of the linear symplectic group. Cubature formulas on the manifold SU(3)/SO(3). Defining a metric on the space of
all symplectic maps.
Understanding, computing, and applying symplectic invariants. For example, given two
multi-particle distributions (in the non-interacting case), when does
there exist a symplectic map that will transform one into the other?
of Synchrotron Radiation
of current understanding of synchrotron-radiation effects using map and
lie-algebraic methods including synchrotron integrals.
derivation of synchrotron radiation effects based on QED.
operator technique: Propagate
distributions using maps for elements and PIC code for space-charge
formation of beam halo.
map and PIC code to calculate the net map that would describe particle
behavior in the beam core.
Multi-particle dynamics is vastly more complicated than
single-particle dynamics and, in comparison to single-particle dynamics
(which, thanks to map methods, is now relatively well understood), almost
nothing is known with certainty or fully understood.
Present Status of Research Work
(Hamiltonian) Orbit Theory
1a, 1b, 1d, and 1f above are essentially complete and have been realized
through 3Őrd order in the computer program MaryLie 3.0, which consists of
some 40,000 lines of code organized into approximately 500 subroutines
and documented in a 900-page MaryLie 3.0 UsersŐ Manual.
Item 1c is nearly complete, and makes possible for the first time
the accurate treatment of real machines with real (non-idealized) beam-line
required theoretical results for item 1e have all been developed and have
been documented in part in the 900-page draft manuscript Lie Methods
for Nonlinear Dynamics with Applications to Accelerator Physics.
The program MaryLie 7.1, which will treat nonlinear effects
through 7th order and also handle all mis-alignment, mis-placement,
and mis-powering errors, is currently under construction.
substantial work has been performed on item 1g. However, there appear to be no major obstacles to the
inclusion of spin based on the map manipulation tools already developed.
substantial work has been done on item 1h. Although this problem is very important, it also
appears to be quite difficult.
work has been carried out on item 1i, but results are so far limited and,
as might be expected, already quite complicated.
work has been carried out on some of the topics listed in item 1j
including symplectic tracking methods, polynomial equivalence classes,
and cubature formulas for the manifold SU(3)/SO(3), and more work on them
appears feasible. Little
work has been done on the remaining topics, and substantial progress for
them awaits a more profound understanding of symplectic geometry.
of Synchrotron Radiation
No substantial work has been performed on item 2a above. However, there appear to be no major
obstacles to the inclusion of synchrotron radiation effects at the classical
and statistical level as currently understood.
With regard to item 2b, preliminary work has been done on
numerically integrating the quantum-mechanical and relativistic equations that
describe the motion of an electron wave packet through a beam-line
element. This work needs to be
extended both analytically and numerically, and subsequently coupling to the
quantized radiation field needs to be included.
3. Multi-particle Effects
The Berkeley/Maryland component of the SciDAC Consortium has
realized item 3a above through 3Őrd order in the program MaryLie/Impact. As MaryLie 7.1 is constructed, it is
expected that a 7th order version of MaryLie/Impact will also be
Controlling and suppressing the development of beam halo is a
fundamental problem in the design of accelerators for the production of intense
beams. There is a partial
understanding of item 3b, but much remains to be learned.
With regard to item 3c, preliminary theoretical work has been
done on the possibility of developing a hybrid map and PIC code to calculate
the net map that describes particle behavior in the beam core. Additional analytical work followed by
numerical experimentation with various beam distributions seems feasible.