# Summer 2013

## Hamiltonian Dynamics

**Announcements**

The course has concluded. Thanks for your interest!

**Exercises**

**Lecture Notes**

The notes are evolving. Current snapshot is here.

**Course description**

(MSc, BSc, WP2; 10 ECTS points; lectures Monday 2 pm - 4 pm, and Wednesday 12:00 pm - 2 pm, SR1 (physics high rise); exercise session Friday 10 am, HS1 (physics high rise). Lecturer: David Gross. Exercises: Johan Aberg)

For an outline, also consult first section of the lecture notes.

This course is concerned with *symplectic geometry* and its role in physics. The main goal is to give a geometric description of Hamiltonian mechanics, but we will take plenty of time to stray away from mechanics and touch on other applications of symplectic methods to physics.

The course consists of three parts:

**1. Symplectic Linear Algebra**. We introduce the basic theory of symplectic vector spaces: forms, the symplectic group, characterization of subspaces, linear symplectic reduction, volume forms. Applications to classical and quantum mechanics are discussed, including: Weyl representation, quantization, Stone-von Neumann Theorem and metaplectic representation, symplectic vector spaces over finite fields and quantum information theory.

**2. Symplectic Manifolds**. Depending on the mathematical education of the audience, we will recap the basic notions of analysis on manifolds: manifolds, vector and tensor fields, differential forms, Cartan and Lie derrivatives. This prepares us for the actual theory: Darboux Theorem, Hamiltonian vector fields, Noether Theorem, symplectic reduction, integrability.

**3. Lie Group Actions and Moment Maps**. The final part deals with Lie group actions on symplectic manifolds. After a recap of Lie group theory, we will introduce the moment map, prove the abelian convexity theorem and consider applications to matrix analysis. Final outlook will be on applications to quantum marginal problems and recent generalizations of the Pauli principle.

**Literature**

V.I. Arnold, *Mathematical Methods of Classical Mechanics*

A. Knauf, *Mathematische Physik: Klassische Mechanik*

G. Rudolph, M. Schmidt, *Differential Geometry and Mathematical Physics*

S. Waldmann, *Poisson-Geometrie und Deformationsquantisierung*

W. Thirring, *Classical Mathematical Physics:Dynamical Systems and Field Theories*

Da Silva, *Lectures on Symplectic Geometry*

## Entropy (Term Paper)

**Course description**

This student seminar will approach the concept of entropy from different perspectives: thermodynamics, information theory, and math.

List of proposed topics available at this url: here.

**Time & Venue**

Wednesdays 2 - 4 pm, venue: Seminarraum Westbau, 2. OG . Organizers: Johan Aberg, Heinz-Peter Breuer, Rafael Chaves, David Gross

Please check regularly for updates of the schedule!

**Schedule**

29.05. *Jayne's principle*, Robert Stierlen

12.06. *Kolmogorov Complexity*, Matthias Dold

19.06. *Negative quantum entropies*, Eduardo Carnio

26.06. *Entropic Forces*, Carolin Willibald

10.07. *Entropy and Non-Locality*, Lukas Luft