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Geometry of torus bundles in integrable Hamiltonian systems

26 September 2008

PhD ceremony: O. Lukina, 13.15 uur, Academiegebouw, Broerstraat 5, Groningen

Thesis: Geometry of torus bundles in integrable Hamiltonian systems

Promotor(s): prof. H.W. Broer

Faculty: Mathematics and Natural Sciences 

 

We are concerned with global properties of Lagrangian bundles, i.e. symplectic n-torus bundles with Lagrangian fibres, as these occur in integrable Hamiltonian systems. Our main interest is in obstructions to triviality and in classification, as well as in manifestations of global invariants in real-world examples of classical and quantum systems. In Chapter 1 we review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. An integrable Hamiltonian system can be viewed geometrically as a torus bundle with a symplectic structure on the total space, whose fibres are Lagrangian, i.e. have tangent spaces which are maximal isotropic with respect to the symplectic structure. From the Liouville- Arnold integrability theorem it follows that an integrable Hamiltonian system is a locally trivial bundle and has symplectic coordinates called action-angle coordinates. A natural question is when such coordinates exist globally. Duistermaat introduced two invariants which measure the failure of these coordinates to exist globally: namely the Lagrange class and the monodromy representation. The monodromy representation is trivial if and only if global action coordinates exist, and their symplectic dual - the angle coordinates - exist precisely when the Lagrange class is trivial as well. It is then natural to ask whether or not the Lagrange class and the monodromy representation completely classify integrable Hamiltonian systems. Nguyen showed that monodromy is completely determined by the integer affine structure on the base space of the bundle, and that bundles with a fixed integer affine structure are classified by the Lagrange class. In Chapter 1 we give new geometric proofs and elucidate certain aspects of the classification results of Duistermaat and Nguyen. In particular, the relation between the integer affine structure and monodromy is explained in detail, and an example of symplectic bundles with the same monodromy but with different integer affine structures is given. An explicit example of topologically equivalent torus bundles with non- isomorphic symplectic structures is constructed, i.e. torus bundles over the same base space with the same Chern class but different Lagrange classes. Chapter 2 deals with manifestations of global invariants of integrable Hamiltonian systems in an example of real-world systems. As was already known, monodromy explains certain phenomena in joint spectra of atoms. We consider integrable approximations of the hydrogen atom in static weak external electric and magnetic fields. This is a continuation of earlier work by Cushman, Zhilinskii, Sadovskii and Efstathiou, who provided the framework to classify all perturbations of the hydrogen atom and introduced the concept of the resonant zone. In Chapter 2 we carry out a detailed study of the 1 : 1 resonant zone, which corresponds to near orthogonal fields. It is demonstrated that for certain domains of parameters the system of the perturbed hydrogen atom exhibits monodromy, and which is visible in the spectrum of the quantized system.

Last modified:15 September 2017 3.38 p.m.

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