Semi-global symplectic invariant of the champagne bottle

2026/05/21 by Ognyan Christov · 1 voice
Physics and Astronomy · #math-ph #nlin.SI

paper · pdf · doi:10.3390/sym18071179

Abstract

We study a two degrees of freedom Hamiltonian system describing the motion of a particle in a potential field of the form of S¹ symmetric double well, namely V = - (x₁² + x₂²) + (x₁² + x₂²)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 Vũ Ng\doc (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).

Citations

• Symplectic invariants of semitoric systems and the inverse problem for quantum systems
• Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points
• Higher index focus-focus singularities in the Jayne-Cummings-Gaudin model : symplectic invariants and monodromy
• Semi-global symplectic invariants of the spherical pendulum
• Vanishing Twist near Focus-Focus Points
• A Cantor set of tori with monodromy near a focus-focus singularity
• Vanishing twist in the Hamiltonian Hopf bifurcation
• On semi-global invariants for focus-focus singularities
• Poincare' normal and renormalized forms
• Convergence versus integrability in Birkhoff normal form
• A note on focus-focus singularities
• On global action‐angle coordinates

Discussions

• Ognyan Christov: Semi-global symplectic invariant of the champagne bottle https://arxiv.org/abs/2605.21985 https://arxiv.org/pdf/2605.21985 https://arxiv.org/html/2605.21985 [bsky, 0 points, 0 comments]

Related

• Semi-global symplectic invariants of the spherical pendulum
• The Classical and Quantum Monodromy of the Champagne bottle potential
• Bohr-Sommerfeld conditions for Integrable Systems with critical manifolds of focus-focus type
• Symplectic Techniques for Semiclassical Completely Integrable Systems
• Hamiltonian Monodromy via spectral Lax pairs