Università Roma Tre

Statistical properties of motion: quasi-periodic motions with irrational frequency vectors. Frequencies of visit: existence and proof that they equal the fraction of phase space occupied, irrespective of the initial datum.
Metric dynamical systems. Birkhoff's theorem (statement).
General definition(s) of ergodic system: absence of non-trivial integral of motions; asymptotic factorization (in average) of the probability of visiting different sets at different times. Mixing systems. An example: the Arnold's cat map. Chaotic systems: Lyapunov exponents, dynamical basis, Oseledec's theorem (statement).

Complements on the theory of the rigid body: integrability of the motion of free rotation of an a-symmetric body around its center of mass. Integrability of the motion of a spinning symmetric body in the gravitational field.

Complements of Hamiltonian mechanics: action-angle variables for 1D conservative motions and for the Kepler's problem. The three-body problem in Keplerian variables. Precession of the perihelium of mercury.

- G. Gallavotti: The elements of mechanics, Springer-Verlag, 1983, available online on:
http://ricerca.mat.uniroma3.it/ipparco/pagine/libri.html]

- G. Gallavotti, F. Bonetto, G. Gentile Aspects of the ergodic, qualitative and statistical theory of motion, Springer-Verlag 2004.

- L.D. Landau, E.M. Lifshitz: Mechanics, Butterworth-Heinemann, 1976.

- V.I. Arnol’d: Mathematical Methods of Classical Mechanics, Springer Graduate Texts in Mathematics.

- G. Gentile: Introduzione ai Sistemi Dinamici: 1 e 2, available online on:
http://www.mat.uniroma3.it/users/gentile/2014-2015/FM410/testo.html