Paper |
Title |
Page |
WEPCH087 |
Normal Form for Beam Physics in Matrix Representation
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2122 |
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- S.N. Andrianov
St. Petersburg State University, Applied Mathematics & Control Processes Faculty, St. Petersburg
- A.N. Chechenin
FZJ, Jülich
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The modeling of long beam evolution dynamics in nonlinear accelerator structures has raised new interest in the effective methods of nonlinear effects calculation. Moreover, it is preferably to use both analytical tools and numerical methods for evolution modeling. Usually the standard numerical methods and computer codes are based on the concept of symplectic transfer maps, whereas the analytical tool is the theory of normal forms. The method of normal forms can be realized in symbolic and numerical modes easily enough. In this paper, we discuss the normal form theory based on the matrix formalism for Lie algebraic tools. This approach allows using well known methods of matrix algebra. This permits to compute necessary matrices step-by-step up to desired order of approximation. This procedure leads to more simple structure of matrix representation for very complicated structure of this map does not allow using this map for practical computing. Therefore, it is necessary to transform this map in more appropriate form. In another words the new matrix representation for the map is particularly simple and has explicit invariants and symmetries.
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WEPCH088 |
High Order Aberration Correction
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2125 |
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- S.N. Andrianov
St. Petersburg State University, Applied Mathematics & Control Processes Faculty, St. Petersburg
- A.N. Chechenin
FZJ, Jülich
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It is known that modern accelerators fall under nonlinear aberrations influence. The most of these aberrations have harmful character, and their effect must be maximally decreased. There are a set of approaches and codes to solving this problem. In this paper, we consider an approach for solving this problem using the matrix formalism for Lie algebraic tools. This formalism allows reducing the starting problem to linear algebraic equations for aberration coefficients, which are elements of corresponding matrices. There are discussed results evaluated using suggested approach and nonlinear programming tools. Some examples of corresponding results are given.
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