Construction of Approximate Invariants Across Nonlinear Resonances
Abstract
Our previous work [Y.~Li, D.~Xu, and Y.~Hao, Phys. Rev. Accel. Beams \textbf{28}, 074001 (2025)] introduced a square-matrix method for constructing approximate invariants recursively, order by order, from the one-turn map of non-integrable Hamiltonian systems. At fourth and higher even orders, the recursive construction is intrinsically non-unique because rotationally invariant polynomials generate a null space in the corresponding homological equations. In the resonant regime, this ambiguity can lead to an incorrect description of the resonance-island topology and is therefore particularly relevant to accelerator applications that rely on stable resonance islands. In this work, we select the null-space contributions by explicitly incorporating the null-space basis vectors and determining their coefficients through an averaging procedure over a prescribed region of interest. The method is demonstrated using the Kobayashi map near a third-order resonance, extending the square-matrix construction to resonant Hamiltonian dynamics.
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BibTeX
@misc{li2026construction,
title = {{Construction of Approximate Invariants Across Nonlinear Resonances}},
author = {Yongjun Li and Derong Xu and Yue Hao and Sergei Nagaitsev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06462},
eprint = {2610.06462},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06462},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Construction of Approximate Invariants Across Nonlinear Resonances
AU - Yongjun Li
AU - Derong Xu
AU - Yue Hao
AU - Sergei Nagaitsev
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06462
UR - https://arxiv.org/abs/2610.06462
AB - Our previous work [Y.~Li, D.~Xu, and Y.~Hao, Phys. Rev. Accel. Beams \textbf{28}, 074001 (2025)] introduced a square-matrix method for constructing approximate invariants recursively, order by order, from the one-turn map of non-integrable Hamiltonian systems. At fourth and higher even orders, the recursive construction is intrinsically non-unique because rotationally invariant polynomials generate a null space in the corresponding homological equations. In the resonant regime, this ambiguity can lead to an incorrect description of the resonance-island topology and is therefore particularly relevant to accelerator applications that rely on stable resonance islands. In this work, we select the null-space contributions by explicitly incorporating the null-space basis vectors and determining their coefficients through an averaging procedure over a prescribed region of interest. The method is demonstrated using the Kobayashi map near a third-order resonance, extending the square-matrix construction to resonant Hamiltonian dynamics.
ER -