First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCN1032593, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2018/19

Information on the course unit
Degree course Second cycle degree in
SC1173, Degree course structure A.Y. 2010/11, A.Y. 2018/19
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Degree course track Common track
Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination MATHEMATICAL PHYSICS
Website of the academic structure
Department of reference Department of Physics and Astronomy
Mandatory attendance
Language of instruction English
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge MASSIMILIANO GUZZO MAT/07

Course unit code Course unit name Teacher in charge Degree course code

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Educational activities in elective or integrative disciplines MAT/07 Mathematical Physics 5.0
Other -- -- 1.0

Course unit organization
Period Second semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Practice 2.0 16 34.0 2
Lecture 4.0 32 68.0 No turn

Start of activities 25/02/2019
End of activities 14/06/2019

Examination board
Board From To Members of the board
7 Commissione Fisica Matematica 18-19 01/10/2018 30/11/2019 GUZZO MASSIMILIANO (Presidente)
FAVRETTI MARCO (Membro Effettivo)

Prerequisites: Linear algebra and calculus with functions of several variables.
Target skills and knowledge: This course provides an introduction to the study of the ordinary and partial differential equations with the modern point of view of dynamical systems. Examples from Astronomy and from Physics will be considered.
Examination methods: Written examination with open questions and exercises on the topics discussed during lectures.
Assessment criteria: The student is asked to use the correct terminology, know the full program of the lectures, be able to link the different topics discussed during lectures and be able to critically apply the methods of dynamical systems.
Course unit contents: The course is given for both students of the Master Degree in Astronomy and Master Degree in Mathematical Engineering. Topics in sections 4) and 5) are only students of the Master Degree in Astronomy, whereas topics in section 6) are only for students of the Master Degree in Mathematical Engineering.

1) Ordinary differential equations: Cauchy theorem, phase-space flow, dependence on the initial conditions; linear equations; phase-portraits, first integrals; equilibrium points; linearizations, stable, center and unstable spaces.
2) Integrable systems: elementary examples from population dynamics, from Mechanics and from Astronomy; integrability of mechanical systems, action-angle variables, examples.
3) Non-integrable Systems: discrete dynamical systems, Poincaré sections; bifurcations, elementary examples. Stable and Unstable manifols, homoclinic chaos; Lyapunov exponents, the forced pendulum and other examples; Center manifolds and partial hyperbolicity. The three body-problem, the Lagrange equilibria,
Lyapunov orbits, the tube manifolds.
4) Linear PDEs of first and second order, well-posed problems,
the vibrating string, 1-dimensional wave equation, normal modes of vibrations, heat equation, Fourier series, 2-dimensional wave equation, Laplace operator and polar coordinates, separation of variables, Bessel functions, eigenfunctions of the Laplacian operator.
5) Laplace operator and spherical coordinates, separation of variables, Legendre polynomials and associate functions, Spherical harmonics, multipole expansions, L2 operator-eigenvalues and eigenfunctions, complete solution of the wave equation in space, Schrodinger polynomials.
6) Examples and Applications: examples of analysis of three and four dimensional systems; limit cycles; the Lorenz system, the three-body problem; examples from fluid dynamics, non autonomous dynamical systems, chaos indicators, Lagrangian Coherent Structures.
Planned learning activities and teaching methods: Classroom lectures and exercises. Lectures are given in English.
Additional notes about suggested reading: "Lecture notes in Mathematical Physics" by M. Guzzo available through the Moodle website of the course on the e-learning platform of the Department of Physics and Astronomy "G. Galilei" (
Textbooks (and optional supplementary readings)

Innovative teaching methods: Teaching and learning strategies
  • Lecturing
  • Loading of files and pages (web pages, Moodle, ...)

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)
  • Mathematica