Abstract

In this paper we consider mathematical and computer modeling of nonlinear dynamics of particle beams in cyclic accelerators in terms of the matrix representation of the corresponding nonlinear differential equations. The proposed approach is different from the usual presentations of non-linear equations in the form of Taylor series. In the paper, we use the coefficients representation in the form of two-dimensional matrices. The similar approach allows us not only to significantly reduce the time spent on modeling beam dynamics but use symbolic mathematics to calculate the necessary two-dimensional matrices. This method demonstrates the effectiveness when solving problems of dynamics problems and optimization of control systems, and for evaluating the influence of various effects on the dynamics of the beam (including taking into account the spin). Using the tools of symbolic computations not only significantly increases the computational efficiency of the method, but also allows you to create databases of "ready-made" transformations (Lego-objects), which greatly simplify the process of modeling particle dynamics. Examples of solving practical problems are given.
Original languageEnglish
Title of host publicationProceedings of the 9th International Particle Accelerator Conference
EditorsShane Koscielniak, Todd Satogata, Volker RW Schaa, Jana Thomson
Place of PublicationGeneva, Switzerland
PublisherJACoW
Pages3554-3556
Number of pages3
ISBN (Print)978-3-95450-184-7
DOIs
Publication statusPublished - Jun 2018

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tensors
matrices
cyclic accelerators
Taylor series
problem solving
particle beams
mathematics
nonlinear equations
differential equations
optimization
coefficients

Cite this

Andrianov, S., Ivanov, A., Kulabukhova, N., Krushinevskii, E., & Sboeva, E. (2018). Dynamic Equations: The Matrix Representation of Beam Dynamic Equations Instead of Tensor Description. In S. Koscielniak, T. Satogata, V. RW. Schaa, & J. Thomson (Eds.), Proceedings of the 9th International Particle Accelerator Conference (pp. 3554-3556). Geneva, Switzerland: JACoW. https://doi.org/10.18429/JACoW-IPAC2018-THPAK134
Andrianov, S. ; Ivanov, A. ; Kulabukhova, N. ; Krushinevskii, E. ; Sboeva, E. / Dynamic Equations: The Matrix Representation of Beam Dynamic Equations Instead of Tensor Description. Proceedings of the 9th International Particle Accelerator Conference. editor / Shane Koscielniak ; Todd Satogata ; Volker RW Schaa ; Jana Thomson. Geneva, Switzerland : JACoW, 2018. pp. 3554-3556
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abstract = "In this paper we consider mathematical and computer modeling of nonlinear dynamics of particle beams in cyclic accelerators in terms of the matrix representation of the corresponding nonlinear differential equations. The proposed approach is different from the usual presentations of non-linear equations in the form of Taylor series. In the paper, we use the coefficients representation in the form of two-dimensional matrices. The similar approach allows us not only to significantly reduce the time spent on modeling beam dynamics but use symbolic mathematics to calculate the necessary two-dimensional matrices. This method demonstrates the effectiveness when solving problems of dynamics problems and optimization of control systems, and for evaluating the influence of various effects on the dynamics of the beam (including taking into account the spin). Using the tools of symbolic computations not only significantly increases the computational efficiency of the method, but also allows you to create databases of {"}ready-made{"} transformations (Lego-objects), which greatly simplify the process of modeling particle dynamics. Examples of solving practical problems are given.",
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Andrianov, S, Ivanov, A, Kulabukhova, N, Krushinevskii, E & Sboeva, E 2018, Dynamic Equations: The Matrix Representation of Beam Dynamic Equations Instead of Tensor Description. in S Koscielniak, T Satogata, VRW Schaa & J Thomson (eds), Proceedings of the 9th International Particle Accelerator Conference. JACoW, Geneva, Switzerland, pp. 3554-3556. https://doi.org/10.18429/JACoW-IPAC2018-THPAK134

Dynamic Equations: The Matrix Representation of Beam Dynamic Equations Instead of Tensor Description. / Andrianov, S.; Ivanov, A.; Kulabukhova, N.; Krushinevskii, E.; Sboeva, E.

Proceedings of the 9th International Particle Accelerator Conference. ed. / Shane Koscielniak; Todd Satogata; Volker RW Schaa; Jana Thomson. Geneva, Switzerland : JACoW, 2018. p. 3554-3556.

Research output

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T1 - Dynamic Equations: The Matrix Representation of Beam Dynamic Equations Instead of Tensor Description

AU - Andrianov, S.

AU - Ivanov, A.

AU - Kulabukhova, N.

AU - Krushinevskii, E.

AU - Sboeva, E.

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N2 - In this paper we consider mathematical and computer modeling of nonlinear dynamics of particle beams in cyclic accelerators in terms of the matrix representation of the corresponding nonlinear differential equations. The proposed approach is different from the usual presentations of non-linear equations in the form of Taylor series. In the paper, we use the coefficients representation in the form of two-dimensional matrices. The similar approach allows us not only to significantly reduce the time spent on modeling beam dynamics but use symbolic mathematics to calculate the necessary two-dimensional matrices. This method demonstrates the effectiveness when solving problems of dynamics problems and optimization of control systems, and for evaluating the influence of various effects on the dynamics of the beam (including taking into account the spin). Using the tools of symbolic computations not only significantly increases the computational efficiency of the method, but also allows you to create databases of "ready-made" transformations (Lego-objects), which greatly simplify the process of modeling particle dynamics. Examples of solving practical problems are given.

AB - In this paper we consider mathematical and computer modeling of nonlinear dynamics of particle beams in cyclic accelerators in terms of the matrix representation of the corresponding nonlinear differential equations. The proposed approach is different from the usual presentations of non-linear equations in the form of Taylor series. In the paper, we use the coefficients representation in the form of two-dimensional matrices. The similar approach allows us not only to significantly reduce the time spent on modeling beam dynamics but use symbolic mathematics to calculate the necessary two-dimensional matrices. This method demonstrates the effectiveness when solving problems of dynamics problems and optimization of control systems, and for evaluating the influence of various effects on the dynamics of the beam (including taking into account the spin). Using the tools of symbolic computations not only significantly increases the computational efficiency of the method, but also allows you to create databases of "ready-made" transformations (Lego-objects), which greatly simplify the process of modeling particle dynamics. Examples of solving practical problems are given.

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BT - Proceedings of the 9th International Particle Accelerator Conference

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Andrianov S, Ivanov A, Kulabukhova N, Krushinevskii E, Sboeva E. Dynamic Equations: The Matrix Representation of Beam Dynamic Equations Instead of Tensor Description. In Koscielniak S, Satogata T, Schaa VRW, Thomson J, editors, Proceedings of the 9th International Particle Accelerator Conference. Geneva, Switzerland: JACoW. 2018. p. 3554-3556 https://doi.org/10.18429/JACoW-IPAC2018-THPAK134