Integrability and beyond

Course guarantor:

Libor Šnobl

Lecturer:

Antonella Marchesiello, Libor Šnobl

Tutor:

Supervisor:

Department of Physics

Synopsis:

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:

Syllabus of lectures:

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. Stackel transformation, relation between the isotropic harmonic oscillator and Coulomb problem

8. Perturbations of integrable and superintegrable systems

9. Normalization and bifurcations around resonances

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] M. Audin: Hamiltonian Systems and Their Integrability. American Mathematical Society, 2008

[2] W. Miller Jr., S. Post and P. Winternitz: Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 423001, 2013

Recommended references:

[3] 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

[4] J. A. Sanders, F. Verhulst, J. Murdock: Averaging Methods in Nonlinear Dynamical Systems, Springer 2007

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans: