Code: 02INB Integrability and beyond
Lecturer: doc. Ing. Libor Šnobl Ph.D. Weekly load: 2P+0C Completion: A
Department: 14102 Credits: 2 Semester: S
Description:
Abstract:
Hamiltonian systems and their integrals of motion. Hamilton-Jacobi equation and separation of variables. Classification of integrable systems with integrals polynomial in momenta. Superintegrability. Perturbative methods in the study of Hamiltonian systems.
Contents:
Outline:
1. Overview of the essentials of differential geometry
2. Symplectic manifolds, Darboux theorem
3. Geometric formulation of Hamiltonian mechanics - Poisson brackets, equations of motion, integrals of motion
4. Liouville & Arnold integrability, action-angle variables
5. Superintegrability, generalized action-angle variables
6. Symplectic reduction
7. Introduction to perturbation theory, Kolmogorov-Arnold-Moser theorem


Recommended literature:
Key references:
[1] W. Thirring, Classical Mathematical Physics: Dynamical Systems and Field Theories, Springer 2003.
[2] M. Audin: Hamiltonian Systems and Their Integrability. American Mathematical Society, 2008.
[3] W. Miller Jr., S. Post and P. Winternitz: Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 423001, 2013.

Recommended references:
[4] E. G. Kalnins, J. M. Kress and W. Miller Jr.: Separation of variables and superintegrability : the symmetry of solvable systems, Institute of Physics Publishing, 2018.
[5] J. A. Sanders, F. Verhulst, J. Murdock: Averaging Methods in Nonlinear Dynamical Systems, Springer 2007.
Keywords:
Integrability, superintegrability, separation of variables

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