Classical mechanics 2

A.Y. 2021/2022
Overall hours
Learning objectives
The course aims to give an introduction to the theory of dynamical systems, with particular emphasis on the coexistence between ordered and chaotic motions in the same system. The simplest methods of perturbation theory and of ergodic theory will be described, and the connections with the dynamical foundations of statistical mechanics illustrated. The course comprises a laboratory for the numerical study of simple models.
Expected learning outcomes
At the end of the course the student will

1) get the notion of the coexistence between ordered and chaotic motions, having observed it in numerical simulations
2) know the notions of hyperbolic fixed point, stable and unstable manifolds, homoclinic points and homoclinic orbits.
3) be able to find, by numerical computations, the stable and unstable manifold for the hyperbolic fixed point of the standard map, and computed the homoclinic orbits.
4) know the statement of the stable manifold theorem.
5) know the model of the torus rotation, understanding the difference between the ration and irrational cases, and know the Arnold's cat map.
6) know the basic notion about the ergodic system, and be able to prove the ergodicity of the irrational torus rotation, and to prove the mixing character of the Arnold's cat map.
7) know the basic notion about the Hamiltonian perturbation theory, and the statement of KAM theorem.
8) be able to prove the stable manifold theorem.
Course syllabus and organization

Single session

Lesson period
Second semester
The course will be delivered entirely remotely in case of restrictions due to Covid-19. The lectures will be offered in virtual classrooms (zoom platform) in synchronous connection.
Course syllabus
-- Introduction to the theory of dynamical systems and to the generic coexistence of ordered and of chaotic motions. 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.

-- Introduction to perturbation theory. The theorem of the mean. The action as an adiabatic invariant. Action-angles variables for completely integrable system and Liouville theorem. The theorem of the mean for system with more then one fast angles.

-- 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.
Prerequisites for admission
Elementary notions on ordinary differential equations, in particular the theorem of existence and uniqueness of the solution for the Cauchy problem.
Elementary notions of Hamiltonian Mechanics: phase space, Hamilton's equations, dynamical variables.

Elementary notions of the theory of measure, Lebesgue integral
Teaching methods
Frontal lectures. There is also a numerical lab, in which the student has to draw the phase portrait for the standard map, the forced pendulum and the Henon-Heiles model.
Teaching Resources
V. I. Arnold and A. Avez, "Ergodic Problems of Classical Mechanics", Addison-Wesley
Carati, Galgani, "Appunti di Meccanica Analitica 2", downloadable from internet
Assessment methods and Criteria
The examination consists in an oral test which focuses on the program topics, in order to ascertain the student understanding of the theory illustrated during in the lectures.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Professor: Carati Andrea
Educational website(s)