Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems
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2025
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| author | Kudryavtseva, Elena A. |
| author_facet | Kudryavtseva, Elena A. |
| contents | In the present paper, we obtain real-analytic symplectic normal forms for integrable Hamiltonian systems with $n$ degrees of freedom near singular points having the type ``universal unfolding of $A_n$ singularity'', $n\ge1$ (local singularities), as well as near compact orbits containing such singular points (semi-local singularities). We also obtain a classification, up to real-analytic symplectic equivalence, of real-analytic Lagrangian foliations in saturated neighborhoods of such singular orbits (semi-global classification). These singularities (local, semi-local and semi-global ones) are structurally stable. It turns out that all integrable systems are symplectically equivalent near their singular points of this type (thus, there are no local symplectic invariants). A complete semi-local (respectively, semi-global) symplectic invariant of the singularity is given by a tuple of $n-1$ (respectively $n-1+\ell$) real-analytic function germs in $n$ variables, where $\ell$ is the number of connected components of the complement of the singular orbit in the fiber. The case $n=1$ corresponds to non-degenerate singularities (of elliptic and hyperbolic types) of one-degree of freedom Hamiltonians; their symplectic classifications were known. The case $n=2$ corresponds to parabolic points, parabolic orbits and cuspidal tori, and the case $n\ge3$ -- to their higher-dimensional analogs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems Kudryavtseva, Elena A. Symplectic Geometry Differential Geometry 53D12, 53D20, 37J35, 70H06 In the present paper, we obtain real-analytic symplectic normal forms for integrable Hamiltonian systems with $n$ degrees of freedom near singular points having the type ``universal unfolding of $A_n$ singularity'', $n\ge1$ (local singularities), as well as near compact orbits containing such singular points (semi-local singularities). We also obtain a classification, up to real-analytic symplectic equivalence, of real-analytic Lagrangian foliations in saturated neighborhoods of such singular orbits (semi-global classification). These singularities (local, semi-local and semi-global ones) are structurally stable. It turns out that all integrable systems are symplectically equivalent near their singular points of this type (thus, there are no local symplectic invariants). A complete semi-local (respectively, semi-global) symplectic invariant of the singularity is given by a tuple of $n-1$ (respectively $n-1+\ell$) real-analytic function germs in $n$ variables, where $\ell$ is the number of connected components of the complement of the singular orbit in the fiber. The case $n=1$ corresponds to non-degenerate singularities (of elliptic and hyperbolic types) of one-degree of freedom Hamiltonians; their symplectic classifications were known. The case $n=2$ corresponds to parabolic points, parabolic orbits and cuspidal tori, and the case $n\ge3$ -- to their higher-dimensional analogs. |
| title | Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems |
| topic | Symplectic Geometry Differential Geometry 53D12, 53D20, 37J35, 70H06 |
| url | https://arxiv.org/abs/2505.12169 |