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


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

Spring 2015

Mechanics and special functions II 



In this semester the series of meetings is devoted to the presentation of selected problems in classical mechanics and celestial mechanics. Special functions appear in  the all problems of modern physics and astronomy, but in today's study programs are generally ignored. Therefore, discussing  the various issues, we will also mention about some classes of special functions.





The second lecture titled 

 The Gauss hypergeometric equation and function


on the day 

02.04.2015, 13:15 in the room 106 building A29, will give 

prof. dr hab. Andrzej Maciejewski


See you there!

About us

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.



We are planning to organize at INSA de Rouen the Workshop "Integrability in Dynamical Systems and Control" on November 14-16, 2012.

  It is going to focus on the following subjects.

    1.  Integrability and non-integrability of dynamical systems.
    2.  Geometry of integrable systems.
    3.  Applications of Differential Galois Theory to integrability.
    4.  Numerical methods in integrability.
    5.  Integrability in optimal control,
    6. and related topics.

The goal is to bring together researchers working on different aspects of integrability problems and to exchange ideas. We would like to invite  to participate in the Workshop and to give a talk. We will be happy to pay the hotel as well as lunches and the conference dinner. We would like to ask to inform us at earliest convenience about the wish to participate in the Workshop.

 Read more about the Workshop on the official website

 Andrzej Maciejewski, Witold Respondek, Vladimir Salnikov

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Phone:      (+48) 683282828


Phone:      (+48) 683282914


Phone:      (+48) 683282918


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.

Links:    Journal,

1.  Yuri N. Fedorov, Andrzej Maciejewski, Maria Przybylska, The Poisson equations in the nonholonomic Suslov problem: integrability, meromorphic and hypergeometric solutions, Nonlinearity,  Vol. 22, no 9, s. 2231--2259, 2009. 

1.  Andrzej Maciejewski, Maria Przybylska, Tomasz Stachowiak, Marek Szydłowski,  Global integrability of cosmological scalar fields, Journal of Physics A : Mathematical and Theoretical, Vol. 41, nr 46, s. [26], 2008.

1.  Maria Przybylska, Finiteness of integrable n-dimensional homogeneous polynimial potentials, Physics Letters A, Vol. 369, no 3, s. 180--187, 2007.

1.  Tomasz Stachowiak, Marek Szydłowski, Andrzej Maciejewski, Nonitegrability of density perturbations in the Friedmann-Robertson-Walker universe,
Journal of Mathematical Physics, Vol. 47, s. 032502-1--032502-11, 2006.

Examples of Poincaré Sections

, , , , , , , , , , , , , , , , ,

Click here [LINK] to see a more detailed analysis.

, , , , , , , , ,

Click here [LINK] to see a more detailed analysis.

, , , , , , , , ,

Click here [LINK] to see a more detailed analysis.