Integrability and beyond

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Physics

Synopsis:

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.

Requirements:

Essential: classical analytical mechanics (canonical momenta, Hamiltons equations of motion etc.).

Recommended: basic knowledge of differential geometry (manifolds, vector fields, differential forms).

Syllabus of lectures:

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

Syllabus of tutorials:

Study Objective:

The students will get a deeper knowledge of the classical Hamiltonian mechanics, better understand the motivation for various standard notions and be able to follow more recent advances in the field, like perturbative methods and superintegrability.

Study materials:

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.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Matematická fyzika (elective course)