Student Seminar: Classical and Quantum Integrable Systems

Contents
1. Liouville Theorem 2
1.1 Dynamical systems of classical mechanics 2
1.2 Harmonic oscillator 5
1.3 The Liouville theorem 7
1.4 Action-angle variables 9
2. Examples of integrable models solved by Liouville theorem 11
2.1 Some general remarks 11
2.2 The Kepler two-body problem 12
2.2.1 Central fields in which all bounded orbits are closed. 15
2.2.2 The Kepler laws 17
2.3 Rigid body 20
2.3.1 Moving coordinate system 20
2.3.2 Rigid bodies 21
2.3.3 Euler’s top 23
2.3.4 On the Jacobi elliptic functions 27
2.3.5 Mathematical pendulum 29
2.4 Systems with closed trajectories 31
3. Lax pairs and classical r-matrix 32
3.1 Lax representation 32
3.2 Lax representation with a spectral parameter 34
3.3 The Zakharov-Shabat construction 36
4. Two-dimensional integrable PDEs 41
4.1 General remarks 42
4.2 Soliton solutions 43
4.2.1 Korteweg-de-Vries cnoidal wave and soliton 43
4.2.2 Sine-Gordon cnoidal wave and soliton 45
4.3 Zero-curvature representation 47
4.4 Local integrals of motion 49
5. Quantum Integrable Systems 55
5.1 Coordinate Bethe Ansatz (CBA) 56
5.2 Algebraic Bethe Ansatz 68
5.3 Nested Bethe Ansatz (to be written) 79
6. Introduction to Lie groups and Lie algebras 80
– 1 –