Geometry of Almost-Conserved Quantities in Symplectic Maps. Part III: Approximate Invariants in Nonlinear Accelerator Systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915964837167104 |
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| author | Zolkin, Tim Nagaitsev, Sergei Morozov, Ivan Kladov, Sergei |
| author_facet | Zolkin, Tim Nagaitsev, Sergei Morozov, Ivan Kladov, Sergei |
| contents | We present a perturbative method for constructing approximate invariants of motion directly from the equations of discrete-time symplectic systems. This framework offers a natural nonlinear extension of the classic Courant-Snyder (CS) theory for systems with one degree of freedom -- a foundational cornerstone in accelerator physics now spanning seven decades and historically focused on linear phenomena. The original CS formalism emerged under conditions where nonlinearities were weak, design goals favored linear motion, and analytical tools -- such as the Kolmogorov-Arnold-Moser (KAM) theory -- had not yet been fully developed. While various normal form methods have been proposed to treat near-integrable dynamics, the approach introduced here stands out for its conceptual transparency, minimal computational overhead, and direct applicability to realistic systems. We demonstrate its power and versatility by applying it to several operational accelerator configurations at the Fermi National Accelerator Laboratory (FermiLab), illustrating how the method enables fast, interpretable diagnostics of nonlinear behavior across a broad range of machine conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_07225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometry of Almost-Conserved Quantities in Symplectic Maps. Part III: Approximate Invariants in Nonlinear Accelerator Systems Zolkin, Tim Nagaitsev, Sergei Morozov, Ivan Kladov, Sergei Chaotic Dynamics Pattern Formation and Solitons Accelerator Physics Applied Physics We present a perturbative method for constructing approximate invariants of motion directly from the equations of discrete-time symplectic systems. This framework offers a natural nonlinear extension of the classic Courant-Snyder (CS) theory for systems with one degree of freedom -- a foundational cornerstone in accelerator physics now spanning seven decades and historically focused on linear phenomena. The original CS formalism emerged under conditions where nonlinearities were weak, design goals favored linear motion, and analytical tools -- such as the Kolmogorov-Arnold-Moser (KAM) theory -- had not yet been fully developed. While various normal form methods have been proposed to treat near-integrable dynamics, the approach introduced here stands out for its conceptual transparency, minimal computational overhead, and direct applicability to realistic systems. We demonstrate its power and versatility by applying it to several operational accelerator configurations at the Fermi National Accelerator Laboratory (FermiLab), illustrating how the method enables fast, interpretable diagnostics of nonlinear behavior across a broad range of machine conditions. |
| title | Geometry of Almost-Conserved Quantities in Symplectic Maps. Part III: Approximate Invariants in Nonlinear Accelerator Systems |
| topic | Chaotic Dynamics Pattern Formation and Solitons Accelerator Physics Applied Physics |
| url | https://arxiv.org/abs/2505.07225 |