Integrability, non-integrability, chaos and control in classical and quantum mechanics

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University of Zielona Góra              Institute of Physics                 Institute of Astronomy       Center for Theoretical Physics, PAS

We are interested in various aspects of solvability and integrability of classical and quantum dynamical systems. The group consists of experienced scientists as well as young researchers, PhD students and undergraduates from the faculties of Physics and of Astronomy. The main subjects of our activity are as follows.

1. Searching for new integrable systems.

2. Finding rigorous proofs of non-integrability and identifying necessary conditions for integrability.

3. Investigating solvability of classical systems in terms of elementary and special functions.

4. Searching for superintegrable and partially integrable classical systems.

5. Stability analysis of classical systems (normal forms etc.).,

6. Studying eigenvalue problems for various quantum systems and searching for solutions in terms of special functions.

7. Analyzing connections between integrability/solvability of quantum systems and their classical counterparts.

8. Studying control problems of classical and quantum systems.

In our research we apply analytical as well as numerical methods to problems. The analytical methods we apply include local forms of solutions around singularities, Birkhoff normal form, solvability analysis of linear equations in classes of entire functions, Liouvillean functions and special functions, monodromy groups, and differential Galois theory. We also integrate systems numerically, make Poincare sections, calculate Lyapunov exponents and use the splitting separatrices method. We analyse systems derived from classical mechanics, including celestial mechanics and cosmology, which may be Hamiltonian, non-Hamiltonian, Poisson and/or non-holonomic.

We kindly invite young people (PhD students and undergraduates) to apply, and we guarantee that the successful applicant will find a wide variety of interesting problems and an opportunity to develop their abilities at scientific work.

## Members

Phone:      (+48) 683282828

Phone:      (+48) 683282914

E-mail:      stachowiak@cft.edu.pl

Phone:      (+48) 683282918

E-mail:      uz88szuminski@gmail.com

Phone:      (+48)722048155

## Our publications

1. G. Duval, A.J. Maciejewski, Integrability of potentials of degree $k\neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity, Discrete and Continuous Dynamical Systems - Series A, 2015, in print

1. A.J. Maciejewski, M. Przybylska, T. Stachowiak, Analytical method of spectra calculations in the Bargmann representation, Phys. Lett. A, Volume 378, Issue 46, 24 October 2014, Pages 3445–3451

1. Wojciech Szumiński, Constrained N-body problems

1.  Andrzej Maciejewski, Maria Przybylska, Tomasz Stachowiak, How to calculate spectra of Rabi and related models

1.  Andrzej Maciejewski, Maria Przybylska, A. V. Tsiganov, On algebraic construction of certain integrable and super-integrable systems, Physica D, Vol. 240, no 18, s. 1426--1448, 2011.

1.   Andrzej Maciejewski, Maria Przybylska, Partial integrability of Hamiltonian systems with homogenous potential, Regular and Chaotic Dynamics, Vol. 15, no 4-5, s. 551--563, 2010.