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Introductory lectures on basic properties of regular and chaotic motion in classical hamiltonian autonomous systems, on the semiclassical quantization of classical chaotic systems and on the spectral properties of random matrix ensembles. Good knowledge of the basis of classical and quantum mechanics is required.
Last update: T_UCJF (21.05.2001)
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The course is concluded with an oral examination. Last update: Stránský Pavel, doc. Mgr., Ph.D. (07.06.2019)
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Classical chaos: Tabor M.: Chaos and Integrability in Nonlinear Dynamics, Wiley, New York (1989). Ott E.: Chaos in Dynamical Systems, Cambridge University Press (1993). Ozorio de Almeida A.M.: Hamiltonian Systems: Chaos and Quantization, Cambridge University Press (1988). Pettini M: Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics, Springer, New York 2007.
Contopoulos G. et al.: Destruction of islands of stability, Journal of Physics A: Mathematical and General 32, 5213 (1999). Meiss J.D.: Symplectic maps, variational principles, and transport, Review of Modern Physics 64, 795 (1992). Skokos Ch.: The Lyapunov Characteristic Exponents and Their Computation, Lecture Notes in Physics 790, 63 (2010).
Quantum chaos: Haake F.: Quantum Signatures of Chaos, Springer (2010). Stöckmann H.-J.: Quantum Chaos: An Introduction, Cambridge University Press (1999). Gutzwiller M.C.: Chaos in Classical and Quantum Mechanics, Springer, New York 1990 Reichl L.E.: The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations (2nd edition), Springer, New York (2004).
Bohigas O.: Random Matrix Theories and Chaotic Dynamics, Les Houches LII, ed. Gianonni M.-J., Voros A., Zinn-Justin J., North-Holland, Amsterdam (1991).
Random matrix theory: Mehta M.L.: Random Matrices (3rd edition), Elsevier (2004) Last update: Stránský Pavel, doc. Mgr., Ph.D. (07.06.2019)
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Theoretical lecture is combined with solving illustrative practical problems on computers ("hands-on") Last update: Stránský Pavel, doc. Mgr., Ph.D. (07.06.2019)
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The examination has an oral form. A student prepares a presentation of an article related to the subject of the course. The exam can be performed at a distance. Last update: Stránský Pavel, doc. Mgr., Ph.D. (28.04.2020)
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Classical Hamiltonian systems. Conditions of integrability. Regularity of motion of integrable systems. Actions and angles, periodical and quasiperiodical trajectories, rational and irrational tori. Poincare surface of section
Perturbations of integrable systems. Convergency of perturbation series. Problem of small denominators. Sufficiently irrational tori. The Kolmogorov-Arnold-Moser theorem. Fate of rational tori. The Birkhoff fixed-point theorem. Stable and instable trajectories. Lyapounov exponents, SALI and GALI methods.
Correspondence between classical and quantum mechanics. Propagators as integrals over paths. Semiclassical quantization of classically chaotic systems. Level density as the Gutzwiller sum over the classical peridic orbits.
Fluctuations of energy levels of quantum systems. Basic fluctuation measures: distribution of nearest-neighbor spacings, rigidity, number variance. Random matrix ensembles. Level fluctuations in GUE and GOE (Gaussian unitary ensemble, Gaussian orthogonal ensemble). Wigner surmise. Brody distribution. Scale invariance of a quantum spectrum and 1/f noise. Peres lattices. Bohigas-Giannoni-Schmit conjecture and its validity. Last update: Stránský Pavel, doc. Mgr., Ph.D. (07.06.2019)
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Classical theoretical mechanics, quantum mechanics and programming at the level of undergraduate courses in physics Last update: Stránský Pavel, doc. Mgr., Ph.D. (07.06.2019)
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