Hamiltonian Monodromy and Morse Theory - Martynchuk, Nikolay; Broer, H. W.; Efstathiou, K. (Communications in Mathematical Physics, 2019)
We show that Hamiltonian monodromy of an integrable two degrees of freedom system with a global circle action can be computed by applying Morse theory to the Hamiltonian of the system. Our proof is based on Takens's index theorem, which specifies how the energy-h Chern number changes when h passes a non-degenerate critical value, and a choice of admissible cycles in Fomenko-Zieschang theory. Connections of our result to some of the existing approaches to monodromy are discussed.
CRIS-Team WoS Importer:2019-10-18

Recent advances in the monodromy theory of integrable Hamiltonian systems - Martynchuk, Nikolay; Broer, H. W.; Efstathiou, K. (Indagationes Mathematicae, 2020)
The notion of monodromy was introduced by J.J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalizations. In particular, we will discuss the monodromy around a focus–focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.
CRIS-Team Scopus Importer:2020-06-02
Keywords: Action–angle coordinates; Hamiltonian system; Liouville integrability; Monodromy; Quantization

Scattering invariants in Euler's two-center problem - Martynchuk, Nikolay; Dullin, Holger; Efstathiou, Konstantinos; Waalkens, Holger (Nonlinearity, 2019)

Topology change of level sets in Morse theory - Knauf, Andreas; Martynchuk, Nikolay (Arkiv For Matematik, 2020)
The classical Morse theory proceeds by considering sublevel sets f−1 (−∞, a] of a Morse function f: M→ℝ, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f−1(a) and give conditions under which the topology of f−1(a) changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level f−1(a) always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M. When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the base space. (Counter-)examples and applications to celestial mechanics are also discussed.
CRIS-Team Scopus Importer:2020-11-20
Keywords: Hamiltonian and celestial mechanics; invariant manifolds; Morse theory; Surgery theory; Vector bundles