Integrability and beyond

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.

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.

Essential: classical analytical mechanics (canonical momenta, Hamilton’s equations of motion etc.), cf. course 02TEF1 Theoretical Physics 1. Recommended: basic knowledge of differential geometry (manifolds, vector fields, differential forms), cf. course 02GMF1 Geometric Methods of Physics 1.

• 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.

Lectures