Events
Department of Mathematics and Statistics
Texas Tech University
 | Wednesday Mar. 5 4:00 Math011
| | Applied Mathematics and Machine Learning TBA Michelle Pantoya Department of Mechanical Engineering, Texas Tech University
|
Abstract.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
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Dynamical systems on 3-manifolds have been paid much attention along the time. In particular, magnetic trajectories are solutions of a second order differential equation (known as the Lorentz equation) and they generalise geodesics. A magnetic field on a Riemannian manifold is defined by a closed 2-form that helps, together with the metric, to define the Lorentz force. On the other hand, magnetic curves derive from the variational problem of the Landau-Hall functional, which is, in the absence of a magnetic field, nothing but the kinetic energy functional.
The dimension 3 is rather special, since it allows us to identify 2-forms with vector fields via the Hodge ⋆ operator and the volume form of the (oriented) manifold. Moreover, in dimension 3, one may define a cross product and therefore, the Lorentz equation may be written in an easier way.
The challenge is to solve the differential equation in order to find an explicit solution, meaning the explicit parametrization for the magnetic trajectories. Nevertheless, this is not always possible and, because of that, it is necessary to understand the behaviour of the solution.
Recent studies are done in 3-dimensional Sasakian manifolds, where the contact 2-form naturally defines a magnetic field. The solutions of the Lorentz equation, usually called contact magnetic trajectories, are often expressed in a concrete parametrization. It can be proved a reduction result for the codimension in a Sasakian space form, that is, essentially, we can reduce the study (of a magnetic curve in a Sasakian space form) to dimension 3. The geometry of magnetic trajectories have been recently studied in the z-sphere, in the Berger 3-sphere, in the Heisenberg group Nil3 and in SL(2,R), respectively.
Another important problem is the existence of closed curves which is a fascinating topic in dynamical systems. Periodic orbits of the contact magnetic fields on the unit 3-sphere and in the Berger 3-sphere were found in the last two decades and conditions for the periodicity have been obtained. A similar result has been recently given; it was proved that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers.
The geometry of contact magnetic curves in SL(2,R) is much more beautiful. More precisely, it can be shown that every contact magnetic trajectory (of charge q) starting at the origin of SL(2,R) with initial velocity X and with charge q is the product of the homogeneous geodesic with initial velocity X and the charged Reeb flow exp(2qs\xi). Similar results are obtained for the Berger spheres as well.
This talk is based on several papers, mainly in collaboration with Prof. Jun-ichi Inoguchi (Japan).
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