C^\∞ symplectic invariants of parabolic orbits and flaps in\n integrable Hamiltonian systems

2021/10/26 by Elena Kudryavtseva, Kudryavtseva, Elena, Nikolay Martynchuk +1
Mathematics · Physics and Astronomy · #37J35 #53D12 #53D20 #70H06 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2110.13758

openalex publication_date 2021/10/26 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

In the present paper, we consider a smooth C^\∞ symplectic\nclassification of Lagrangian fibrations near cusp singularities, parabolic\norbits and cuspidal tori. We show that for these singularities as well as for\nan arrangement of singularities known as a flap, which arises in the integrable\nsubcritical Hamiltonian Hopf bifurcation, the action variables form a complete\nset of C^\∞ symplectic invariants. We also give a symplectic\nclassification for parabolic orbits in the real-analytic case. Namely, we prove\nthat a complete symplectic invariant in this case is given by a real-analytic\nfunction germ in two variables. Additionally, we construct several symplectic\nnormal forms in the C^\∞ and/or real-analytic categories, including\nreal-analytic right and right-left symplectic normal forms for parabolic\norbits.\n

Citations

• On global action‐angle coordinates

Related

• Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom
• Symplectic classification for universal unfoldings of A_n singularities in integrable systems
• On the symplectic geometry of A_k singularities
• On the adjoint action of the group of symplectic diffeomorphisms
• On the C^∞ closing lemma for Hamiltonian flows on symplectic 4-manifolds