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arxiv: 2505.07225 · v3 · submitted 2025-05-12 · 🌊 nlin.CD · nlin.PS· physics.acc-ph· physics.app-ph

Geometry of Almost-Conserved Quantities in Symplectic Maps. Part III: Approximate Invariants in Nonlinear Accelerator Systems

Tim Zolkin , Sergei Nagaitsev , Ivan Morozov , Sergei Kladov This is my paper

Pith reviewed 2026-05-22 16:42 UTC · model grok-4.3

classification 🌊 nlin.CD nlin.PSphysics.acc-phphysics.app-ph
keywords approximate invariantssymplectic mapsCourant-Snyder theorynonlinear dynamicsaccelerator physicsperturbative methodsdiscrete time systems
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The pith

A perturbative method constructs approximate invariants directly from discrete-time symplectic system equations, extending Courant-Snyder theory nonlinearly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a perturbative method for building approximate invariants of motion from the equations of discrete-time symplectic systems. This provides a nonlinear extension to the Courant-Snyder theory that has been key in accelerator physics for linear motion over seven decades. The method is applied to several operational accelerator setups, showing its use for diagnosing nonlinear effects with minimal computation. Readers would find it useful for gaining insight into near-integrable nonlinear dynamics in practical systems.

Core claim

The central claim is that approximate invariants can be constructed perturbatively and directly from the discrete-time symplectic equations, offering a transparent nonlinear generalization of the classic Courant-Snyder invariants for one-degree-of-freedom systems, as demonstrated in FermiLab accelerator configurations.

What carries the argument

Perturbative expansion around the linear Courant-Snyder solution applied to the map equations to produce approximate invariants.

If this is right

  • The method allows direct applicability to realistic accelerator systems with weak nonlinearities.
  • It provides fast, interpretable diagnostics of nonlinear behavior across various machine conditions.
  • It offers conceptual transparency and minimal computational overhead relative to normal form methods.
  • Applications to operational configurations at Fermi National Accelerator Laboratory illustrate its versatility.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the method works as described, it could be extended to analyze resonance effects or stability in other symplectic map systems beyond accelerators.
  • Connections to KAM theory suggest that these approximate invariants could quantify the persistence of tori under perturbations.
  • Future work might involve comparing the approximate invariants against numerical tracking results to quantify accuracy in specific lattices.

Load-bearing premise

The underlying dynamics remain close enough to the linear integrable solution that perturbative corrections yield useful approximations.

What would settle it

If in an actual nonlinear accelerator the approximate invariants constructed this way fail to remain nearly constant over many turns, beyond the expected small variations from the perturbative order, the claim would be falsified.

Figures

Figures reproduced from arXiv: 2505.07225 by Ivan Morozov, Sergei Kladov, Sergei Nagaitsev, Tim Zolkin.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison between numerically computed trajectories (black dots) and constant level sets of the approximate [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a perturbative method for constructing approximate invariants of motion directly from the equations of discrete-time symplectic systems. This is positioned as a nonlinear extension of the classic Courant-Snyder theory for one-degree-of-freedom systems, with applications demonstrated on operational accelerator configurations at FermiLab to provide diagnostics of nonlinear behavior.

Significance. If the constructed invariants can be shown to remain usefully conserved with quantifiable error bounds in the relevant amplitude and tune regimes, the framework would supply a transparent, low-overhead alternative to normal-form techniques for near-integrable symplectic maps. The direct construction from the map equations (with no fitted parameters) and explicit ties to KAM ideas constitute a clear methodological strength that could aid fast diagnostics in accelerator lattices.

major comments (2)
  1. Abstract and application sections: the assertion of successful application to FermiLab configurations is not accompanied by quantitative validation metrics, error estimates, or direct comparisons against the linear Courant-Snyder invariant. Without these, it is impossible to determine whether the approximate invariants improve conservation properties by a factor that survives realistic lattice errors and resonance proximity.
  2. Perturbative construction (method description): no a-priori estimate (e.g., majorant-series or Nekhoroshev-type bound) is supplied for the radius of convergence or the size of the remainder after the displayed truncation orders. This is load-bearing for the central claim that the truncated invariant remains usefully conserved under the full nonlinear map in the cited operational regimes.
minor comments (1)
  1. Notation for the perturbative orders and the resulting approximate invariant should be introduced with a single consistent symbol set early in the text to avoid ambiguity when comparing to the linear CS case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major points raised below and indicate where revisions will be made to improve clarity and substantiation of the claims.

read point-by-point responses

  1. Referee: Abstract and application sections: the assertion of successful application to FermiLab configurations is not accompanied by quantitative validation metrics, error estimates, or direct comparisons against the linear Courant-Snyder invariant. Without these, it is impossible to determine whether the approximate invariants improve conservation properties by a factor that survives realistic lattice errors and resonance proximity.

    Authors: We agree that the current presentation of the FermiLab applications would benefit from more explicit quantitative support. In the revised manuscript we will add direct comparisons of the variation of the constructed approximate invariants against the linear Courant-Snyder invariant over representative turn numbers, together with tabulated relative-error measures and assessments under small lattice perturbations. These additions will appear in the application sections and will be summarized concisely in the abstract so that readers can judge the practical improvement in conservation. revision: yes

  2. Referee: Perturbative construction (method description): no a-priori estimate (e.g., majorant-series or Nekhoroshev-type bound) is supplied for the radius of convergence or the size of the remainder after the displayed truncation orders. This is load-bearing for the central claim that the truncated invariant remains usefully conserved under the full nonlinear map in the cited operational regimes.

    Authors: The referee is correct that no rigorous a-priori bound on the remainder or radius of convergence is provided. Deriving Nekhoroshev-type or majorant-series estimates for this particular perturbative construction on general symplectic maps is a substantial separate undertaking that lies outside the scope of the present work. The manuscript instead demonstrates utility through explicit construction and numerical checks on the cited operational lattices. In revision we will expand the method section to state the truncation orders employed, supply heuristic remainder estimates based on the magnitude of the nonlinear coefficients, and clarify that applicability in the reported regimes rests on these numerical validations rather than on general analytic bounds. revision: partial

Circularity Check

0 steps flagged

No significant circularity; direct perturbative construction from map equations

full rationale

The paper presents a perturbative method for constructing approximate invariants directly from the discrete-time symplectic map equations as a nonlinear extension of Courant-Snyder theory. The abstract and description emphasize conceptual transparency and direct applicability to the map without reference to fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the core claim to prior unverified work by the same authors. The derivation chain is self-contained against the input map equations and the stated assumption of weak nonlinearities, with no exhibited reduction of any prediction or invariant to its own inputs by construction. This is the expected honest non-finding for a method claimed to be built from first principles on the system equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the symplectic maps are near-integrable so that a perturbative series around the linear solution remains meaningful; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption The discrete-time symplectic map is near-integrable with sufficiently weak nonlinearities for a perturbative expansion to be useful.
    Invoked via references to the historical limitations of Courant-Snyder theory and to KAM theory in the abstract.

pith-pipeline@v0.9.0 · 5724 in / 1272 out tokens · 78329 ms · 2026-05-22T16:42:37.119836+00:00 · methodology

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 1 internal anchor

  1. [1]

    Theory of the alternating-gradient synchrotron,

    Ernest D. Courant and Hartland S. Snyder, “Theory of the alternating-gradient synchrotron,” Annals of Physics 3, 1–48 (1958)

    work page 1958

  2. [2]

    (World Sci- entific Publishing Company, 2018)

    Shyh-Yuan Lee,Accelerator Physics, 4th ed. (World Sci- entific Publishing Company, 2018)

    work page 2018

  3. [3]

    XXIX, 1021

    Helmut Wiedemann,Particle accelerator physics, 4th ed., Graduate Texts in Physics (Springer Cham, 2015) pp. XXIX, 1021

    work page 2015

  4. [4]

    Nonlinear accelerator lat- tices with one and two analytic invariants,

    V. Danilov and S. Nagaitsev, “Nonlinear accelerator lat- tices with one and two analytic invariants,” Phys. Rev. ST Accel. Beams13, 084002 (2010)

    work page 2010

  5. [5]

    Experimental demonstration of optical stochastic cooling,

    Jonathan Jarvis, V. Lebedev, A. Romanov, D. Broem- melsiek, K. Carlson, S. Chattopadhyay, A. Dick, D. Ed- strom, I. Lobach, S. Nagaitsev,et al., “Experimental demonstration of optical stochastic cooling,” Nature608, 287–292 (2022)

    work page 2022

  6. [6]

    Essay: A path for the construction of a muon collider,

    Diktys Stratakis, “Essay: A path for the construction of a muon collider,” Phys. Rev. Lett.134, 160001 (2025)

    work page 2025

  7. [7]

    Fu- ture Circular Collider Feasibility Study Report Volume 1: Physics and Experiments,

    Wolfgang Bartmann, Jean-Paul Burnet, Christian Carli, Antoine Chance, Paolo Craievich, Massimo Giovannozzi, Christophe Grojean, Johannes Gutleber, Klaus Hanke, Andre Henriques, Patrick Janot, Carlos Lourenco, Michelangelo Mangano, Thomas Otto, John Howard Poole, Srini Rajagopalan, Tor Raubenheimer, Ezio Tode- sco, Timothy Paul Watson, and Guy Wilkinson, “...

    work page 2025

  8. [8]

    Future Circular Collider Feasibility Study Report Volume 2: Ac- celerators, technical infrastructure and safety,

    Michael Benedikt, Wolfgang Bartmann, Jean-Paul Bur- net, Christian Carli, Antoine Chance, Paolo Craievich, Massimo Giovannozzi, Christophe Grojean, Johannes Gutleber, Klaus Hanke, Andre Henriques, Patrick Janot, Carlos Lourenco, Michelangelo Mangano, Thomas Otto, John Howard Poole, Srini Rajagopalan, Tor Rauben- heimer, Ezio Todesco, Luisa Ulrici, Timothy...

    work page 2025

  9. [9]

    Future Circular Collider Feasibility Study Report Volume 3: Civil Engineering, Implementation and Sus- tainability,

    Johannes Gutleber, Luisa Ulrici, and Timothy Paul Wat- son, “Future Circular Collider Feasibility Study Report Volume 3: Civil Engineering, Implementation and Sus- tainability,” (2025)

    work page 2025

  10. [10]

    Geometry of almost-conserved quanti- ties in symplectic maps,

    Tim Zolkin, Sergei Nagaitsev, Ivan Morozov, and Sergei Kladov, “Geometry of almost-conserved quanti- ties in symplectic maps,” Chaos, Solitons & Fractals208, 118059 (2026)

    work page 2026

  11. [11]

    Chaos and time- reversal symmetry. order and chaos in reversible dynam- ical systems,

    J.A.G. Roberts and G.R.W. Quispel, “Chaos and time- reversal symmetry. order and chaos in reversible dynam- ical systems,” Physics Reports216, 63–177 (1992)

    work page 1992

  12. [12]

    Resonant normal forms, interpolat- ing Hamiltonians and stability analysis of area preserv- ing maps,

    A. Bazzani, M. Giovannozzi, G. Servizi, E. Todesco, and G. Turchetti, “Resonant normal forms, interpolat- ing Hamiltonians and stability analysis of area preserv- ing maps,” Physica D: Nonlinear Phenomena64, 66–97 (1993). 7

    work page 1993

  13. [13]

    Betatron fre- quency and the Poincar´ e rotation number,

    Sergei Nagaitsev and Timofey Zolkin, “Betatron fre- quency and the Poincar´ e rotation number,” Phys. Rev. Accel. Beams23, 054001 (2020)

    work page 2020

  14. [14]

    Erratum: Beta- tron frequency and the Poincar´ e rotation number [Phys. Rev. Accel. Beams 23, 054001 (2020)],

    Sergei Nagaitsev and Timofey Zolkin, “Erratum: Beta- tron frequency and the Poincar´ e rotation number [Phys. Rev. Accel. Beams 23, 054001 (2020)],” Phys. Rev. Accel. Beams29, 029901 (2026)

    work page 2020

  15. [15]

    Michelotti Leo,Intermediate classical dynamics with ap- plications to beam physics(Wiley, New York, 1995) pp. 284–299

    work page 1995

  16. [16]

    8 (Hardwood Academic / CRC Press, Amsterdam, The Netherlands, 1998) pp

    ´Etienne Forest,Beam Dynamics: A New Attitude and Framework, The Physics and Technology of Particle and Photon Beams, Vol. 8 (Hardwood Academic / CRC Press, Amsterdam, The Netherlands, 1998) pp. 138–141

    work page 1998

  17. [17]

    Chapter 7 - repetitive systems,

    Berz Martin, “Chapter 7 - repetitive systems,” inMod- ern Map Methods in Particle Beam Physics, Advances in Imaging and Electron Physics, Vol. 108, edited by Peter Hawkes (Elsevier, 1999) pp. 270–288

    work page 1999

  18. [18]

    Analysis of nonlinear dynamics by square matrix method,

    Li Hua Yu, “Analysis of nonlinear dynamics by square matrix method,” Phys. Rev. Accel. Beams20, 034001 (2017)

    work page 2017

  19. [19]

    Ivan Morozov, “square,”https://github.com/ i-a-morozov/square(2024)

    work page 2024

  20. [20]

    Jacques Gareyte, Jean-Pierre Koutchouk, and F Rug- giero,Landau damping dynamic aperture and octupole in LHC, Tech. Rep. (CERN, Geneva, 1997) revised version number 1 submitted on 2003-08-21 14:12:02

    work page 1997

  21. [21]

    Landau damping of beam instabilities by electron lenses,

    V. Shiltsev, Y. Alexahin, A. Burov, and A. Vali- shev, “Landau damping of beam instabilities by electron lenses,” Phys. Rev. Lett.119, 134802 (2017)

    work page 2017

  22. [22]

    Dy- namics of McMillan mappings I. McMillan multipoles,

    Tim Zolkin, Sergei Nagaitsev, and Ivan Morozov, “Dy- namics of McMillan mappings I. McMillan multipoles,” Physica D: Nonlinear Phenomena476, 134620 (2025)

    work page 2025

  23. [23]

    Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity

    Tim Zolkin, Sergei Nagaitsev, Ivan Morozov, and Sergei Kladov, “Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity,” (2026), arXiv:2410.10380 [accepted toNonlinear Dyn]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  24. [24]

    Betatron motion with coupling of horizontal and vertical degrees of free- dom,

    Valeri A. Lebedev and S.A. Bogacz, “Betatron motion with coupling of horizontal and vertical degrees of free- dom,” Journal of Instrumentation5, P10010 (2010)

    work page 2010

  25. [25]

    Symmetry properties of a symplectic transport matrix and Twiss parameterization of a fully coupled motion,

    Sergei Glukhov, “Symmetry properties of a symplectic transport matrix and Twiss parameterization of a fully coupled motion,” Phys. Rev. Accel. Beams28, 084001 (2025)

    work page 2025

  26. [26]

    Extracting dy- namical frequencies from invariants of motion in finite- dimensional nonlinear integrable systems,

    Chad E. Mitchell, Robert D. Ryne, Kilean Hwang, Sergei Nagaitsev, and Timofey Zolkin, “Extracting dy- namical frequencies from invariants of motion in finite- dimensional nonlinear integrable systems,” Phys. Rev. E 103, 062216 (2021)

    work page 2021

  27. [27]

    Dynamics of McMillan mappings II. Axially symmetric map,

    Tim Zolkin, Brandon Cathey, and Sergei Nagaitsev, “Dynamics of McMillan mappings II. Axially symmetric map,” Nonlinear Dynamics113, 20253–20283 (2025)

    work page 2025

  28. [28]

    Marylie 3.0 - A program for nonlineapr analysis of accelerator and beamline lattices,

    Alex J. Dragt, Liam M. Healy, Filippo Neri, Robert D. Ryne, David R. Douglas, and Etienne Forest, “Marylie 3.0 - A program for nonlineapr analysis of accelerator and beamline lattices,” IEEE Transactions on Nuclear Science32, 2311–2313 (1985)

    work page 1985

  29. [29]

    Dragt,Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics(University of Maryland, Center for Theoretical Physics, Department of Physics, College Park, 1997)

    Alex J. Dragt,Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics(University of Maryland, Center for Theoretical Physics, Department of Physics, College Park, 1997)

    work page 1997

  30. [30]

    108, edited by Peter Hawkes and Martin Berz (Elsevier,

    Martin Berz, inModern map methods in particle beam physics, Advances in Imaging and Electron Physics, Vol. 108, edited by Peter Hawkes and Martin Berz (Elsevier,

    work page