**Maryland Dynamical Systems and Accelerator Theory **

**http://www.physics.umd.edu/dsat/**

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.

In
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.

** **

- Single-particle (Hamiltonian) Orbit Theory
- Linear approximation including coupling between all degrees of freedom.
- Nonlinear properties of machines with idealized beam-line elements.
- Maps for real (non-idealized) beam-line elements based on 3-dimensional field data and including all nonlinear multipole and fringe-field effects.
- Concepts and tools for analysis of nonlinear behavior including normal-form methods.
- Efficient code for computing and analyzing nonlinear behavior of real machines with complicated lattices.
- Behavior of multi-particle distributions and distribution functions in absence of multi-particle interactions. Construction of moment invariants for the symplectic group.
- Treatment of spin, including maps for all standard (idealized and real) elements and Siberian snakes.
- Direct prediction of long-term behavior (e.g., dynamic aperture) based on properties of one-turn map.
- Possible construction of generalized moment invariants for nonlinear symplectic maps.
- Understanding 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?

- Inclusion of Synchrotron Radiation
- Implementation of current understanding of synchrotron-radiation effects using map and lie-algebraic methods including synchrotron integrals.
- First-principles derivation of synchrotron radiation effects based on QED.

- Multi-particle Effects
- Split operator technique: Propagate distributions using maps for elements and PIC code for space-charge effects.
- Understand formation of beam halo.
- Hybrid map and PIC code to calculate the net map that would describe particle behavior in the beam core.
- Improved understanding. 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.

- Single-particle (Hamiltonian) Orbit Theory
- Items
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 elements. - The
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 7^{th}order and also handle all mis-alignment, mis-placement, and mis-powering errors, is currently under construction. - No 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.
- No substantial work has been done on item 1h. Although this problem is very important, it also appears to be quite difficult.
- Some work has been carried out on item 1i, but results are so far limited and, as might be expected, already quite complicated.
- Some 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.

- Inclusion of Synchrotron Radiation

a. 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.

b. 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

a.
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 7^{th} order version of MaryLie/Impact will also be
developed.

b. 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.

c. 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.