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Regular and Chaotic Dynamics

, Volume 21, Issue 6, pp 697–706 | Cite as

Knauf’s degree and monodromy in planar potential scattering

  • Nikolay MartynchukEmail author
  • Holger Waalkens
On the 70th Birthday of Nikolai N. Nekhoroshev Special Memorial Issue. Part 1
  • 32 Downloads

Abstract

We consider Hamiltonian systems on (T*ℝ2, dqdp) defined by a Hamiltonian function H of the “classical” form H = p 2/2 + V(q). A reasonable decay assumption V(q) → 0, ‖q‖ → ∞, allows one to compare a given distribution of initial conditions at t = −∞ with their final distribution at t = +∞. To describe this Knauf introduced a topological invariant deg(E), which, for a nontrapping energy E > 0, is given by the degree of the scattering map. For rotationally symmetric potentials V(q) = 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 deg(E) and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree 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 

MSC2010 numbers

37J35 70F99 70H05 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands

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