Nikolay Martynchuk
Publications:
Kudryavtseva E. A., Martynchuk N. N.
Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori
2021, vol. 26, no. 6, pp. 732741
Abstract
We show that every parabolic orbit of a twodegreeoffreedom 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 nonsymplectic 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.

Martynchuk N. N., Waalkens H.
Knauf’s Degree and Monodromy in Planar Potential Scattering
2016, vol. 21, no. 6, pp. 697706
Abstract
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.
