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Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom

Alexey Bolsinov, Lorenzo Guglielmi, Elena Kudryavtseva

TL;DR

The paper develops a symplectic-invariant framework for parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. It first establishes a canonical parabolic model $H = \pm(x^2 + y^3 + b(\lambda) y + a(\lambda))$, $F = \lambda$, and then analyzes the symplectic structure near a parabolic orbit via a tubular $(x,y,\lambda,\varphi)$-description, reducing to a $1$-degree-of-freedom problem on each $\lambda$-slice. It derives intrinsic invariants from action variables (and associated period data) that classify parabolic singularities, both in the single-parameter and parametric contexts, and extends these ideas to semi-local invariants of cuspidal tori: two fibrations with matching cusp bifurcation diagrams and identical action data are fiberwise symplectomorphic. The results rely on complexification, versal deformation arguments, and a careful analysis of period mappings, yielding necessary and sufficient conditions for local and semi-local symplectic equivalence and highlighting the canonical affine structure of the base as the fundamental semi-local invariant. Overall, the work provides a systematic, invariant-based approach to degenerate singularities in low-dimensional integrable systems with potential generalization to more degrees of freedom.

Abstract

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom

TL;DR

The paper develops a symplectic-invariant framework for parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. It first establishes a canonical parabolic model , , and then analyzes the symplectic structure near a parabolic orbit via a tubular -description, reducing to a -degree-of-freedom problem on each -slice. It derives intrinsic invariants from action variables (and associated period data) that classify parabolic singularities, both in the single-parameter and parametric contexts, and extends these ideas to semi-local invariants of cuspidal tori: two fibrations with matching cusp bifurcation diagrams and identical action data are fiberwise symplectomorphic. The results rely on complexification, versal deformation arguments, and a careful analysis of period mappings, yielding necessary and sufficient conditions for local and semi-local symplectic equivalence and highlighting the canonical affine structure of the base as the fundamental semi-local invariant. Overall, the work provides a systematic, invariant-based approach to degenerate singularities in low-dimensional integrable systems with potential generalization to more degrees of freedom.

Abstract

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.

Paper Structure

This paper contains 7 sections, 34 theorems, 128 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definition of parabolic singularities. Canonical form with no symplectic structure
  3. Description of a neighborhood of a parabolic orbit with symplectic structure
  4. The case $\lambda=0$, one-degree of freedom problem
  5. Parametric version
  6. Semi-local invariants of cusp singularities
  7. Appendix

Key Result

Proposition 2.1

Locally in a neighborhood of $P$ there exist a transformation and a local coordinate system $x,y,\lambda, \varphi$ such that $(x,y,\lambda,\varphi)|_P=(0,0,0,0)$ and

Figures (3)

  • Figure 1: Two cross-sections $N_1,N_2$ to the fibration defined by $H=y^3-x^2$
  • Figure 2: Two cross-sections $\mathcal{N}_1,\mathcal{N}_2$ to the fibration near a parabolic orbit
  • Figure 3: Singular fibration near a cuspidal torus

Theorems & Definitions (67)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Remark 2.3
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 57 more