Students

Here you can find the list of students who defended their bachelor/diploma theses under my supervision.

 

Aleš Flandera, 2016

Investigation of chaotic dynamical systems using the methods of Riemannian geometry

Institute of theoretical physics, Faculty of Mathematics and Physics, Charles University in Prague

Abstract. While the formalism of isolated horizons is known for some time, only quite recently the near horizon solution of Einstein’s equations has been found in the Bondi-like coordinates by Krishnan in 2012. In this framework, the spacetime is regarded as the characteristic initial value problem with the initial data given on the horizon and another null hypersurface. It is not clear, however, what initial data reproduce the simplest physically relevant black hole solution, namely that of Kerr–Newman which describes stationary, axisymmetric black hole with charge. Moreover, Krishnan’s construction employs the non-twisting null geodesic congruence and the tetrad which is parallelly propagated along this congruence. While the existence of such tetrad can be easily established in general, its explicit form can be very difficult to find and, in fact it has not been provided for the Kerr–Newman metric. The goal of this thesis was to fill this gap and provide a full description of the Kerr–Newman metric in the framework of isolated horizons. In the theoretical part of the thesis we review the spinor and Newman–Penrose formalism, basic geometry of isolated horizons and then present our results. Thesis is complemented by several appendices.

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Table of contents

  1. 1. Tetrad formalism
  2. 2. Non-expanding horizons
  3. 3. Weakly isolated horizons
  4. 4. Non-twisting null tetrad in Kerr-Newman spacetime
  5. 5. Newman-Penrose formalism
  6. 6. Series
  7. 7. Mathematica source code

 

arXiv:gr-qc/1611.02215

David Matejov, 2016

Investigation of chaotic dynamical systems using the methods of Riemannian geometry

Institute of theoretical physics, Faculty of Mathematics and Physics, Charles University in Prague

Abstract. In this thesis we investigate a chaos in dynamical systems described by the Hamilton function using a new geometric method. At first, necessary definitions and terms are introduced, like dynamical systems, Lyapunov exponents and Poncaré sections. Subsequently we deal with the new geometric method (Horowitz et al., 2007), which predicts the stability of a system under consideration. An extensive part is devoted to the general relativity and spinor formalism, the Newman–Penrose formalism and the theory of optical scalars. In the computational part of the thesis, we investigate the pp-waves and chaotic behaviour of geodesics in this class of space-times. All calculations are done in the Python programming language, so we the chapter on numerical calculations is included.

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Table of contents

  1. 1. Introduction
  2. 2. Dynamical systems
  3. 3. Numerical methods
  4. 4. Riemann geometry of Hamiltonian systems
  5. 5. General theory of relativity
  6. 6. Analysis of the Hénon-Heiles potential
  7. 7. Conclusions
  8. 8. Python codes

 

Lukáš Holka, 2014

Spinorial techniques for constructing quasi-local quantities in general relativity

Institute of theoretical physics, Faculty of Mathematics and Physics, Charles University in Prague

Abstract. It is possible to define reasonable global mass for asymptotically flat spacetimes. In this work we compute the Bondi mass of weakly asymptotically flat spacetime that contains interacting scalar and electromagnetic fields. We then obtain the Bondi mass-loss formula and show that it is negatively semi-definite. These results are derived with the help of spinorial techniques which we introduce in the first part of this thesis, which also contains brief review of several other constructions of energy in general relativity.

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Table of contents

  1. 1. Spinors in general relativity
  2. 2. Spinor analysis
  3. 3. GHP formalism
  4. 4. Twistor equation and twistors
  5. 5. The geometry of spacelike 2-surfaces
  6. 6. Penrose's mass
  7. 7. Mass in general relativity
  8. 8. Einstein-electro-scalar equations
  9. 9. Electro-scalar fields in the Newman-Penrose formalism
  10. 10. Asymptotic solution of Einstein-electro-scalar equations
  11. 11. On the Bondi mass of Maxwell-Klein-Gordon spacetimes

 

The last chapter is the copy of the paper published in General Relativity and Gravitation

Petr Veselý, 2013

Selected problems in computational fluid dynamics

Department of Air Transport, Faculty of Transportation Sciences, Czech Technical University in Prague

Abstract. This batchelor thesis deals with the study of two-dimensional stationary potential flow and with the modelling of incompressible flow past the obstacles. In the theoretical part of the thesis we introduce the notion of two-dimensional flow and basic mathematical apparatus, in particular the theory of functions of complex variable and the description of the flow using the complex potential. Next we explain the method of vortex panels in CFD (computational fluid dynamics) in detail and apply it to a problem of the flow past an airfoil of arbitrary shape. This method is implemented in the modelling and computational software Mathematica.

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Table of contents

  1. 1. Two dimensional flow
  2. 2. Flow past the cylinder
  3. 3. Airfoils
  4. 4. Vortex panel method
  5. 5. Results
  6. 6. Conclusions
  7. Appendices
  8. A. The source code
  9. B. Demonstrations

 

This thesis won the prize "Best bachelor thesis" of Faculty of transportation sciences, CTU in Prague

Miroslav Vaniš, 2013

Mathematical modelling of selected problems in transportation in Java

Department of Applied Mathematics, Faculty of Transportation Sciences, Czech Technical University in Prague

Abstract. In this bachelor thesis we have studied selected mathematical models developed to mimic the behavior of the real traffic flow. In the theoretical part we introduced some basic ideas related to the models of the traffic flow. We introduce both macroscopic and microscopic models, then we restrict our attention to the latter and briefly review existing microscopic models. Next we introduce the elementary theory of dynamical systems and discuss the stability analysis by linearization. Finally we present numerical methods for solving ordinary differential equations (Euler method, Runge-Kutta methods) and present the so-called Intelligent Driver Model (IDM) in detail. In the practical part of the thesis we implement the IDM model in computer language Java. Our application, being the most important result of the thesis, solves dynamical equations of IDM model and represents the results both in the graphic form and in the form of data files. Moreover, we have extended basic IDM model by overtaking model known as MOBIL. At the end of the thesis, after explaining main features and algorithms used in the application, we present full source code and a number of results obtained.

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Table of contents

  1. Introduction
  2. 1. Macroscopic models
  3. 2. Microscopic models
  4. 3. Dynamical systems
  5. 4. Numerical methods
  6. 5. Intelligent driver model
  7. 6. Description of the program
  8. 7. The source code
  9. 8. Simulations
  10. 9. Conclusions

 

The thesis is written in he Czech language.

Miroslav Tomášik, 2012

Geodesics in helically symetric spacetimes

Institute of theoretical physics, Faculty of Mathematics and Physics, Charles University in Prague

Abstract. In this bachelor thesis we investigate geodesics in helically symmetric spacetimes in the frame- work of linearized Einstein’s gravity. Work is an extension of paper by Bičák, Scholtz and Bohata which is under preparation. First we introduce standard numerical methods for solv- ing systems of ordinary differential equations. Next we present helically symmetric solution of linearized Einstein’s equations and numerical code solving the geodesic equation on given back- ground. We discuss conditions of existence of helically symmetric solution and finally we present selected results obtained by numerical simulations. We give present few particular examples of geodesics, selected phase portraits obtained by the method of the Lyapunovov exponents and visualize the causal structure of helically symmetric spacetime.

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Table of contents

  1. Introduction
  2. 1. Numerical methods
  3. 2. Newtonian solution
  4. 3. Helically symmetric soluton
  5. 4. Implementation
  6. 5. Results
  7. 6. Conclusions
  8. 7. Epilogue

 


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