pith. sign in

arxiv: 2605.21985 · v1 · pith:2IBOCWYUnew · submitted 2026-05-21 · 🌊 nlin.SI · math-ph· math.MP

Semi-global symplectic invariant of the champagne bottle

Ognyan Christov This is my paper

Pith reviewed 2026-05-22 02:30 UTC · model grok-4.3

classification 🌊 nlin.SI math-phmath.MP
keywords champagne bottle potentialfocus-focus singularitysemi-global symplectic invariantBirkhoff normal formintegrable Hamiltonian systemsmonodromytwo degrees of freedom
0
0 comments X

The pith

The champagne bottle Hamiltonian yields an explicit semi-global symplectic invariant near its focus-focus singularity.

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

The paper studies the integrable two-degree-of-freedom system given by the champagne bottle potential, which has a non-trivial monodromy that rules out global action variables. The authors focus on the neighborhood of the focus-focus equilibrium and compute the Birkhoff normal form to extract the nontrivial action. From this they derive the semi-global symplectic invariant in the sense of Vũ Ngọc. The result supplies one of the few concrete examples of this invariant and is compared directly to the spherical pendulum case.

Core claim

For the champagne bottle Hamiltonian the Birkhoff normal form is calculated near the focus-focus singularity, the nontrivial action is identified, and the semi-global symplectic invariant is obtained as the function that encodes the symplectic geometry in a punctured neighborhood of the singularity.

What carries the argument

The Birkhoff normal form near the focus-focus point, which supplies the nontrivial action that remains well-defined despite the monodromy.

If this is right

  • The explicit invariant distinguishes the champagne bottle from other integrable systems sharing the same local singularity type.
  • Local dynamics near the equilibrium can be described without requiring global action coordinates.
  • The comparison with the spherical pendulum invariant highlights system-specific features of the monodromy.
  • The same normal-form procedure applies to other members of the class of integrable systems with non-trivial monodromy.

Where Pith is reading between the lines

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

  • The method could be used to construct a table of semi-global invariants for other known integrable Hamiltonians with focus-focus points.
  • The explicit form opens the possibility of studying semiclassical approximations or quantization conditions that respect the monodromy.
  • Small perturbations of the potential could be tested to see how the invariant changes and whether it remains stable under deformation.

Load-bearing premise

The champagne bottle system is completely integrable, so that a semi-global symplectic invariant near the focus-focus singularity is well-defined once the Birkhoff normal form is known.

What would settle it

An independent calculation of the action variable near the focus-focus equilibrium that produces a different functional dependence for the invariant would falsify the result.

Figures

Figures reproduced from arXiv: 2605.21985 by Ognyan Christov.

Figure 1
Figure 1. Figure 1: The set Ur of regular values of the momentum map. Let us choose a basis of the homology group H1(Th,j2 , Z) with the following representatives: for γ1 we fix pφ and φ and let r, pr make one circle on the curve 1 2  p 2 r + p 2 φ r 2  + r 4 − r 2 = h; 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The β - cycle (blue) and the vanishing α - cycle (red) Before stating the next lemma we recall that the functions F(x; κ) = Z x 0 dt p (1 − t 2 )(1 − κ 2 t 2 ) = Z θ 0 dt p 1 − κ 2 sin2 t =: F(θ; κ), E(x; κ) = Z x 0 r 1 − κ 2 t 2 1 − t 2 dt = Z θ 0 p 1 − κ 2 sin2 tdt =: E(θ; κ), Π(n; x; κ) = Z x 0 dt (1 − nt2 ) p (1 − t 2 )(1 − κ 2 t 2 ) = Z θ 0 dt (1 − n sin2 t) p 1 − κ 2 sin2 t =: Π(n; θ, κ) are called i… view at source ↗
Figure 3
Figure 3. Figure 3: The singular fibre (pinched torus) at the focus-focus point. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
read the original abstract

We study a two degrees of freedom Hamiltonian system describing the motion of a particle in a potential field of the form of $S^1$ symmetric double well, namely $V = - (x_1^2 + x_2^2) + (x_1^2 + x_2^2)^2$, known also as a champagne bottle potential. This system is completely integrable. The champagne bottle is the simplest member of a class of integrable systems that have no global action variables due to a non-trivial monodromy, Bates (1991). Beyond that, the geometric and dynamical properties of the system near the equilibrium are of primary interest. We calculate the Birkhoff normal form and the nontrivial action near the focus-focus singularity and obtain the semi-global symplectic invariant near focus-focus point, which is introduced by V\~{u} Ng\d{o}c (2003). Examples of such calculations are still few. We compare our result with the semi-global symplectic invariant of the spherical pendulum, calculated by Dullin (2013).

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

0 major / 2 minor

Summary. The manuscript investigates the two-degree-of-freedom champagne bottle Hamiltonian with potential V = -(x₁² + x₂²) + (x₁² + x₂²)². The system is completely integrable but possesses non-trivial monodromy, precluding global action variables. The authors compute the Birkhoff normal form near the focus-focus equilibrium at the origin, extract the associated nontrivial action, and obtain the semi-global symplectic invariant in the sense of Vũ Ngọc (2003). The result is compared with the corresponding invariant for the spherical pendulum computed by Dullin (2013).

Significance. If the explicit calculation is correct, the work supplies a concrete additional example of a semi-global symplectic invariant for a focus-focus singularity in an integrable system with monodromy. The champagne bottle is a standard, simple model in this class, and the direct comparison with the spherical pendulum case supplies a useful external consistency check. Such explicit computations remain scarce and can serve as benchmarks for further theoretical developments.

minor comments (2)
  1. The abstract states that the Birkhoff normal form and semi-global invariant are obtained but supplies neither the leading coefficients nor the explicit functional form of the invariant; including these would allow immediate assessment of the result.
  2. A brief, self-contained recap of the Vũ Ngọc construction (including the role of the nontrivial action) in the introduction or a dedicated preliminary section would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately reflects the content and goals of the work, including the computation of the Birkhoff normal form near the focus-focus equilibrium and the extraction of the semi-global symplectic invariant for the champagne bottle system, along with the comparison to the spherical pendulum.

Circularity Check

0 steps flagged

No significant circularity; concrete calculation follows external framework

full rationale

The paper performs an explicit computation of the Birkhoff normal form and the associated semi-global symplectic invariant for the champagne-bottle Hamiltonian near its focus-focus equilibrium. This follows the construction introduced in the external reference Vũ Ngọc (2003) and uses the known integrability and monodromy properties established independently in Bates (1991). The result is compared against the independent calculation for the spherical pendulum in Dullin (2013). No self-citations appear in the load-bearing steps, no parameters are fitted to a subset and then relabeled as predictions, and no ansatz or uniqueness claim is smuggled in via prior work by the same author. The derivation is therefore self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The calculation is presumed to rest on standard symplectic geometry and the definition of the semi-global invariant from the cited reference.

axioms (1)
  • domain assumption The champagne bottle system is completely integrable.
    Stated in the abstract as the starting point for the normal-form calculation.

pith-pipeline@v0.9.0 · 5708 in / 1104 out tokens · 49073 ms · 2026-05-22T02:30:33.081828+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Abraham, J

    R. Abraham, J. E. Marsden, Foundations of Mechanics, Benjamin, Reading, 1978

    work page 1978

  2. [2]

    Alonso, H

    J. Alonso, H. R. Dullin, S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, J. Geom. Phys.140, 131–151, 2019

    work page 2019

  3. [3]

    Alonso, On the symplectic invariants of semitoric systems, PhD Thesis, University of Antwerp, 2019

    J. Alonso, On the symplectic invariants of semitoric systems, PhD Thesis, University of Antwerp, 2019

    work page 2019

  4. [4]

    V. I. Arnold, Mathematical methods of Classical Mechanics, Springer, Berlin, 1978

    work page 1978

  5. [5]

    Babelon and B

    O. Babelon and B. Dou¸ cot. Higher index focus-focus singularities in the Jaynes- Cummings-Gaudin model: symplectic invariants and monodromy. J. Geom. Phys.,87, 3—29, 2015

    work page 2015

  6. [6]

    Bates, Monodromy in the champagne bottle, J

    L. Bates, Monodromy in the champagne bottle, J. Appl. Math. Phys. (ZAMP),42, 837–847, 1991

    work page 1991

  7. [7]

    Bates, M

    L. Bates, M. Zou, Degeneration of Hamiltonian monodromy cycles, Nonlinearity6, 313– 335, 1993

    work page 1993

  8. [8]

    A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems: geometry, topology, classification, Chapman & Hall, 2004

    work page 2004

  9. [9]

    A. V. Bolsinov, A. A. Oshemkov, Singularities in integrable Hamiltonian systems, Topo- logical methods in the theory of integrable systems, Cambridge Sci. Publ., 1-67, 2006

    work page 2006

  10. [10]

    P. Byrd, M. Friedman, Handbook of elliptic integrals for engineers and physicists, Springer, Berlin, 1954

    work page 1954

  11. [11]

    Cayley, An Elementary Treatise on Elliptic Functions, Second Ed, Dover, New York, 1961

    A. Cayley, An Elementary Treatise on Elliptic Functions, Second Ed, Dover, New York, 1961

    work page 1961

  12. [12]

    M. S. Child, Quantum States in a champagne bottle, J. Phys. A.,31(2), 657–670, 1998

    work page 1998

  13. [13]

    Deprit, Canonical transformation depending on small parameter, Celest

    A. Deprit, Canonical transformation depending on small parameter, Celest. Mech.72, 173–179, 1969

    work page 1969

  14. [14]

    Dufour, P

    J. Dufour, P. Molino, and A. Toulet. Classification des syst` emes int´ egrables en dimension 2 et invariants des mod` eles de Fomenko. C. R. Acad. Sci. Paris S´ er. I Math.,318(10), 949—952, 1994

    work page 1994

  15. [15]

    Dullin, San V˜ u Ngo.c, Vanishing twist near focus-focus points, Nonlinearity17, 1777–1785, 2004

    H.R. Dullin, San V˜ u Ngo.c, Vanishing twist near focus-focus points, Nonlinearity17, 1777–1785, 2004

    work page 2004

  16. [16]

    Dullin, A

    H.R. Dullin, A. V. Ivanov, Vanishing twist in the Hamiltonian Hopf bifurcation, Physica D201, 27–44, 2005. 19

    work page 2005

  17. [17]

    Dullin, Semi-global symplectic invariants of the spherical pendulum, J

    H.R. Dullin, Semi-global symplectic invariants of the spherical pendulum, J. Differential Equations,254, 2942–2963, 2013

    work page 2013

  18. [18]

    J. J. Duistermaat, On global action-angle coordinates, Comm. Pure and Appl. Math. 33, 687–706, 1980

    work page 1980

  19. [19]

    L. H. Eliasson, Hamiltonian systems with Poisson commuting integrals, PhD Thesis, University of Stockholm, 1984

    work page 1984

  20. [20]

    Gaeta, Poincar´ e normal and renormalized forms, Acta Appl

    G. Gaeta, Poincar´ e normal and renormalized forms, Acta Appl. Math.,70(1-3), 113–131, 2002

    work page 2002

  21. [21]

    Georgiev, Perturbations in a champagne bottle, Ann

    G. Georgiev, Perturbations in a champagne bottle, Ann. de l’Universite de Sofia ”St. Kliment Ohridski”, Faculte de Math´ ematiques et Informatique, Livre 1- Math´ ematiques et Mecanique, Tome89, 141–157, 1995

    work page 1995

  22. [22]

    Horozov, Perturbations of the spherical pendulum and Abelian integrals, J

    E. Horozov, Perturbations of the spherical pendulum and Abelian integrals, J. Reine Angew. Math.408, 114—135, 1990

    work page 1990

  23. [23]

    Horozov, On the isoenergetical nondegeneracy of the spherical pendulum, Phys

    E. Horozov, On the isoenergetical nondegeneracy of the spherical pendulum, Phys. Lett. A173(3), 279—283, 1993

    work page 1993

  24. [24]

    Ito, Convergence of Birkhoff normal forms for integrable systems, Comment

    H. Ito, Convergence of Birkhoff normal forms for integrable systems, Comment. Math. Helv.64(3), 412–461, 1989

    work page 1989

  25. [25]

    Meyer, G

    K. Meyer, G. Hall, D. Offin, Introduction to Hamiltonian Dynamical Systems and the N- Body Problem, Appl. Math. Sci. 90, Second Edition, Springer Science+Business Media, LLC 2000

    work page 2000

  26. [26]

    San V˜ u Ngo.c, On semi-global invariants for focus-focus singularities, Topology42(2), 365–380, 2003

    work page 2003

  27. [27]

    Pelayo, T

    A. Pelayo, T. S. Ratiu, and S. V˜ u Ngo.c, The affine invariant of proper semitoric integrable systems. Nonlinearity,30(11), 3993—4028, 2017

    work page 2017

  28. [28]

    Rink, A Cantor set of tori with monodromy near a focus-focus singularity, Nonlinearity 17, 347–356, 2004

    B. Rink, A Cantor set of tori with monodromy near a focus-focus singularity, Nonlinearity 17, 347–356, 2004

    work page 2004

  29. [29]

    Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer

    J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math.58(1), 141–163, 1936

    work page 1936

  30. [30]

    N. T. Zung, A topological classification of integrable hamiltonian systems. In R. Brouzet (Ed.), S´ eminaire Gaston Darboux de g´ eometrie et topologie diff´ erentielle, pages 43—54. Universit´ e Montpellier II, 1994-1995

    work page 1994

  31. [31]

    N. T. Zung, Kolmogorov condition for integrable systems with focus-focus singularities, Phys. Lett. A,215, 40–44, 1996. 20

    work page 1996

  32. [32]

    N. T. Zung, A note on focus-focus singularities, Diff. Geometry and its Aplications,7, 123–130, 1997. 21

    work page 1997