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arXiv · 2106.04838

Existence of a smooth Hamiltonian circle action near parabolic orbits

Elena Kudryavtseva·Nikolay Martynchuk
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Abstract

We show that every parabolic orbit of a two-degree of freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the integrals of motion is Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.

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Elena Kudryavtseva, Nikolay Martynchuk. 2021-06-09. Existence of a smooth Hamiltonian circle action near parabolic orbits. https://doi.org/10.1134/s1560354721060101

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