Integrable Systems
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WMMA03705 
Vaknaam  Integrable Systems 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Integrable Systems  
Leerdoelen  At the end of the course, the student is able to: 1. explain what it means for a given Hamiltonian system to be integrable in the Liouville sense. Reproduce concrete examples of integrable systems discussed in this course and verify the integrability using one of the methods discussed (group invariance, separation of variables, AKS scheme). 2. perform a qualitative topological analysis of one of the integrable 2degree of freedom systems discussed in the course: construct the bifurcation diagram, identify the singularities and their type, specify the bifurcations of Liouville tori. 3. describe the obstructions to the existence of global actionangle variables in integrable systems. Prove the nonexistence of global action variables for integrable systems with focusfocus points, such as the spherical pendulum and Lagrange top. 4. reproduce the definition of a Lax pair and Lax pair with spectral parameter. Explain the difference between the existence of a Lax pair, the general notion of Liouville integrability, and algebraic integrability. Give examples of algebraically integrable systems and compute their Lax pairs. 5. give examples of integrable manybody systems and show how they arise via symplectic reduction. Reproduce the integrable PDEs discussed in this course and solve them via the inverse scattering method. Explain a connection between manybody systems and integrable PDEs (particlesoliton duality). 6. explain what is integrability in quantum mechanics and its relation to the geometry of classically integrable systems. Give examples of quantum spin chains and outline the quantum inverse scattering method. 

Omschrijving  Integrable systems form an especially important class of "exactly solvable" dynamical systems without dissipation. Such systems naturally appear in a variety of problems of (classical, quantum, and statistical) physics and mechanics, several famous examples being the Kepler problem, the (spherical) pendulum, integrable spinning tops, integrable manybody systems (CalogeroMoser, spin chains), and integrable PDEs (the KdV, NSE, sineGordon equations...) In this course, we will discuss integrability from several different perspectives and outline their connections to other fields in dynamical systems, (symplectic) geometry, and mathematical physics. We will start with the classical notion of integrability due to J. Liouville and explain how the exactly solvable models mentioned above are integrable in this precise sense. We will then discuss qualitative features of (finitedimensional) integrable systems, including their typical singularities, topological, and symplectic invariants. Our prime examples will be the spherical pendulum and the integrable spinning tops. In the next part of the course, we will reconsider the above integrable models in the more algebraic context of Lax pairs and the AdlerKostantSymes scheme  a powerful method for proving integrability of a given Hamiltonian system. This will pave our way towards a more detailed treatment of manybody systems of CalogeroMoser and Toda and finally to integrable PDEs and the inverse scattering transform. The course will be concluded with a panoramic overview of quantum integrability, including a discussion of quantum spin chains and the quantum inverse scattering method. 

Uren per week  
Onderwijsvorm 
Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)
(Being present at 70% of the lectures and at the presentations of homework assignments is compulsory for passing the course.) 

Toetsvorm 
Opdracht (AST)
(The final grade is computed based on the average grade A of 3 homework assignments and active participation T in the tutorials as follows: 0.9 A+ 0.1 T. Solutions to homework assignments are presented by the students in class before the grades are finalised.) 

Vaksoort  master  
Coördinator  Dr. T.F. Görbe  
Docent(en)  Dr. T.F. Görbe ,dr. N. Martynchuk  
Verplichte literatuur 


Entreevoorwaarden  The course assumes a good knowledge of the material covered in Bachelor mathematics courses, including basic dynamical systems, differential forms, topological spaces, groups and their actions. A background in Lie algebras and Hamiltonian mechanics will be helpful, but is not required.  
Opmerkingen  
Opgenomen in 
