Integrability and beyond

Garant předmětu:

Libor Šnobl

Lecturer:

Antonella Marchesiello, Libor Šnobl

Tutor:

Antonella Marchesiello, Libor Šnobl

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, Hamilton’s equations of motion etc.), cf. course 02TEF1.

Recommended: basic knowledge of differential geometry (manifolds, vector fields, differential forms), cf. course 02GMF1.

Syllabus of lectures:

Outline:

1. Review of Hamiltonian mechanics - Poisson brackets, equations of motion, integrals of motion, Hamilton-Jacobi equation

2. Separation of variables in Hamilton-Jacobi equation, action-angle variables

3. Levi-Civita condition for separability, separation in orthogonal coordinate systems, relation of separability to the existence of integrals of motion

4. Conditions determining integrals of motion polynomial in the momenta in the Euclidean space

5. Classification of 2D & 3D quadratically integrable systems

6. Superintegrability, polynomial algebras of integrals

7. Perturbations of integrable and superintegrable systems

8. Normalization and bifurcations around resonances

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:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Matematická fyzika (elective course)