Università Roma Tre

Systems of linear differential equations with constant coefficients.

The damped and forces harmonic oscillator with a general periodic force. Fourier's theorem for periodic functions.

Parametric resonance: evolution operator, Liouville's theorem, stability criterium. Perturbative computation of the evolution operator, convergence of the series. Construction of the stbaility boundary around the resonances; stability diagram.

Systems of coupled harmonic oscillators with Dirichlet boundary conditions: solution of the problem. Continuum (macroscopic) limit: rescaling of distances, masses, elastic constants. Limiting equation: the wave equation. Uniform convergence of the (Fourier's) series of the solution in the continuum limit: exchange of limits. Unicity of the solution of the wave equation: conservation of the energy.

Introduction to classical scattering theory. Differential scattering cross section. Rutherford's formula.

Hidden conserved quantities in the Kepler's problem and in the 3D harmonic oscillator. Bertrand's theorem (statement).

- 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