### Completely Integrable System I: Mathematical Methods of Classical Mechanics (Summer Semester 2021)

- Lecturer: JProf. Dr. Xian Liao
- Classes: Seminar (0173500)
- Weekly hours: 2

This Seminar will be held online via Microsoft Teams. Please register by sending me (xian.liao@kit.edu) an email with your KIT Email address, your study stage (e.g. in which semester of your Bachelor/Master study) and the possibly interested topic(s) until 22.03.2021. The registers will be added to the Microsoft Team Group, where all the informations concerning this seminar will be announced.

The first meeting to distribute the topics is scheduled at 14:00 on Thursday 25.03.2021 via Zoom (Meeting-ID: 573 264 9920).

We recommend to do a presentation in English, but German is also acceptable.

Schedule | |||
---|---|---|---|

Seminar: | Tuesday 14:00-15:30 | Online Microsoft Teams | Begin: 27.4.2021, End: 23.7.2021 |

We propose a series of seminars starting from Summer Semester 2021, each of which would be independent and self-contained. The ultimate goal is to study some interesting completely integrable Hamiltoni-an Partial Differential Equations (PDEs) which have Lax-pair formulations and are completely integrable by means of inverse scattering methods. Typical examples include the Korteweg-de Vries equation (KdV) and the one-dimensional cubic nonlinear Schrödinger equations (NLS).

To have a clear geometric picture of these, in this semester SS21 we shall start by studying mathematical methods in classical mechanics. Our plan is to go through the Lagrangian formalism of classical mechanics, while emphasizing on some concrete classical integrable systems such as central force problems and integrable rigid body motions. We shall follow the book “Mathematical Methods of Classical Mechanics” by V.I. Arnold 1 and we propose the following topics:

1. Experimental facts. (Sections 1-3 1)

2. Systems with one/two degree(s) of freedom. (Sections 4-5 1)

3. Central fields problems. (Sections 6-9 1)

4. Conservation laws and similarity methods. (Sections 10-11 1)

5. Variational principles. (Sections 12-14 1)

6. Liouville’s theorem. (Sections 15-16 1)

7. Lagrangian dynamical systems. (Sections 17-19 1)

8. Noether’s theorem and D’Alembert’s principle. (Sections 20-21 1)

9. Linearization and small oscillations. (Sections 22-23 1)

10. Characteristic frequencies and parametric resonance. (Sections 24-25 1)

11. Inertial forces and Coriolis force. (Sections 26-27 1)

12. Rigid bodies. (Sections 28-29 1)

13. Lagrange’s top. (Sections 30-31 1)

Prerequisites: Analysis I-II, Linear Algebra.

This seminar is proper for Bachelor students.

# References

1. V.I. Arnold, Mathematical Methods of Classical Mechanics. Second Edition. Springer. 1989.