Classical mechanics 2

A.Y. 2018/2019
Lesson for
Overall hours
Learning objectives
Il corso e' tenuto congiuntamente da Luigi Galgani e da Andrea Carati

Obiettivi per fisici (o matematici)
Fornire una introduzione alla teoria qualitativa dei sistemi dinamici, alla transizione tra ordine e caos,
alla teoria delle perturbazioni e alla teoria ergodica,
particolarmente in connessione ai fondamenti dinamici della meccanica statistica

Course structure and Syllabus

Active edition
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Professor: Carati Andrea
Chaper 1. Introduction to the theory of dynamical systems and to the generic coexistence of ordered and of chaotic motions, in the example of the standard map.
The last theorem of Poincare' in the Arnold version. The appearing of chaos through the hyperbolic fixed points and the homoclinic orbits. Proof of the stable manifold theorem.

Chapter 2. Introduction to perturbation theory. The theorem of the mean. The action as an adiabatic invariant.

Chapter 3. Introduction to ergodic theory. Physical motivation for studying time averages. Role of probability on the initial data, Liouville's theorem and the return theorem of Poincare', the ergodic theorem of von Neumann and its physical interpretation.
The approach to equilibrium and the mixing properties of flows; decay of time-correlations. Macroscopic irreversibility through a microscopic reversibility. The example of the specific heat, and ``deduction'' of the second law of thermodynamics through the fluctuation dissipation theorem. Thermodynamics as a large deviations theory; justification of the canonical distribution.
Lesson period
Second semester
Lesson period
Second semester
Assessment methods
Assessment result
voto verbalizzato in trentesimi