Colloquium Mathematics: Dr. K. Efstathiou, University of Groningen

Title: Nontrivial Topology and Symmetry in Integrable Hamiltonian Systems

Abstract:
Action-angle coordinates offer a complete description of integrable Hamiltonian systems; except when they do not. It was understood in the ‘80s that whether action-angle coordinates can be globally defined in a system depends on its geometric (topological and symplectic) properties [1]. The most fundamental, topological, obstruction to the existence of action-angle coordinates is Hamiltonian monodromy. Soon after the introduction of monodromy for integrable Hamiltonian systems, it was understood that it has has nontrivial implications for the structure of the spectrum of the corresponding quantum system. In particular, monodromy has been found in atomic and molecular spectra of fundamental systems such as the hydrogen atom in crossed electric and magnetic fields, and the water molecule.

What is the source of monodromy in an integrable Hamiltonian system? If the system has a local or global circle symmetry then a deep recent result states that monodromy is completely determined through an analysis of the points that remain fixed under the symmetry action [2] while the particular form of the Hamiltonian plays a secondary role. This result, with appropriate modifications, further allows to understand the source of a generalized type of monodromy, known as fractional Hamiltonian monodromy [3, 4, 5]. In this talk we show how monodromy (standard and fractional) manifests in specific examples of (classical and quantum) physical systems and how to compute monodromy by carefully analyzing sets where the symmetry of the system has non-trivial isotropy.

References

[1] J. J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics 33, no. 6, 687–706 (1980).

[2] K. Efstathiou and N. Martynchuk, Monodromy of Hamiltonian systems with complexity 1 torus actions, Journal of Geometry and Physics, 115:104–115 (2017).

[3] N.N. Nekhoroshev, D.A. Sadovskií, and B.I. Zhilinskií, Fractional Hamiltonian monodromy, Annales Henri Poincaré 7, 1099-1211 (2006).

[4] K. Efstathiou and H.W. Broer, Uncovering fractional monodromy, Communications in Mathematical Physics 324, no. 2, 549-588 (2013).

[5] N. Martynchuk and K. Efstathiou, Fractional Hamiltonian monodromy and parallel transport on Seifert manifolds https://arxiv.org/abs/1609.07003.

Colloquium coordinators are Prof.dr. A.J. van der Schaft ( a.j.van.der.schaft rug.nl ), Dr. A.V. Kiselev (e-mail: a.v.kiselev rug.nl )

http://www.rug.nl/research/jbi/news/colloquia/mathematics-colloquia/