Nikolay Martynchuk

Nijenborgh 9, 9747 AG Groningen
University of Groningen


Kudryavtseva E. A., Martynchuk N. N.
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. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.
Keywords: Liouville integrability, parabolic orbit, circle action, structural stability, normal forms
Citation: Kudryavtseva E. A., Martynchuk N. N.,  Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 732-741
Martynchuk N. N., Waalkens H.
Knauf’s Degree and Monodromy in Planar Potential Scattering
2016, vol. 21, no. 6, pp.  697-706
We consider Hamiltonian systems on $(T^{*}\mathbb R^2, dq \wedge dp)$ defined by a Hamiltonian function $H$ of the “classical” form $H = p^2/2 + V (q)$. A reasonable decay assumption $V(q) \to 0, \, \|q\| \to \infty$, allows one to compare a given distribution of initial conditions at $t = −\infty$ with their final distribution at $t = +\infty$. To describe this Knauf introduced a topological invariant $\text{deg}(E)$, which, for a nontrapping energy $E > 0$, is given by the degree of the scattering map. For rotationally symmetric potentials $V = W(\|q\|)$, scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree $\text{deg}(E)$ and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree $\text{deg}(E)$, which appears when the nontrapping energy $E$ goes from low to high values.
Keywords: Hamiltonian system, Liouville integrability, nontrapping degree of scattering, scattering monodromy
Citation: Martynchuk N. N., Waalkens H.,  Knauf’s Degree and Monodromy in Planar Potential Scattering, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 697-706

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