First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCP4065477, A.A. 2019/20

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

Information on the course unit
Degree course First cycle degree in
SC1158, Degree course structure A.Y. 2014/15, A.Y. 2019/20
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Number of ECTS credits allocated 9.0
Type of assessment Mark
Course unit English denomination FUNDAMENTS OF MATHEMATICAL PHYSICS
Website of the academic structure
Department of reference Department of Physics and Astronomy
Mandatory attendance No
Language of instruction Italian
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 FRANCESCO FASSO' MAT/07

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Educational activities in elective or integrative disciplines MAT/07 Mathematical Physics 9.0

Course unit organization
Period Second semester
Year 2nd Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Lecture 9.0 72 153.0 No turn

Start of activities 02/03/2020
End of activities 12/06/2020
Show course schedule 2019/20 Reg.2014 course timetable

Examination board
Board From To Members of the board
3 Istituzioni di Fisica Matematica 01/10/2018 30/11/2019 FASSO' FRANCESCO (Presidente)
PONNO ANTONIO (Membro Effettivo)
ROSSI PAOLO (Supplente)

Prerequisites: Mathematical Analysis 1,2,3. Geometry. General Physics 1.
Target skills and knowledge: Thorough comprehension of the Lagrangian formulation of classical mechanics, in a rigorous mathematical framework. Bases of the theory of dynamical systems and of the Hamiltonian formulation of classical mechanics.
Ability of modelling, analyzing and studying the dynamics of mechanical systems formed by point masses, either free or constrained, using the appropriate mathematical technics. Ability of working at a theoretical level with the formalism of Lagrangian Mechanics.
Examination methods: The examination consists of a written test on both theory and exercises.
This type of examination is meant to stimulate the student to work in parallel, synergically, on theory and exercises. Moreover, it makes it possible to evaluate the global comprehension of the matter reached by the student.
Assessment criteria: The exams aims at establishing the knowledge of the subject and the capability of solving pertinent exercises.
The evaluation will focus on the student's knowledge and comprehension of the matter, particularly through the completeness and precision of the the answers, the methodological rigor, and the clarity of the exposition.
Course unit contents: The aim of the course is the formulation of Classical Mechanics in a rigorous mathematical framework, and the introduction of the Lagrangian and Hamiltonian formalism.

1. Qualitative theory of ordinary differential equations: Flow of a differential equation. First integrals, Lie derivative. Linearization at an equilibrium. Phase portraits of linear differential equations and of conservative systems in the plane. Stability of equilibria; Lyapunov theorems. Invariant submanifolds and ordinary differential equations on submanifolds.

2. Constrained systems: Holonomic constraints; configuration manifolds and Lagrangian coordinates. Indeal contraints. Kinetic energy, forces and potential energy in Lagrangian coordinates. Lagrange equations: deduction and normal form.

3. Lagrangian Mechanics: Invariance of Lagrange equations. Equivalent Lagrangians. Jacobi integral; conservation of energy. Velocity-dependent potentials: electromagnetic forces in the Lagrangian formalism. Equilibria and stability: Lagrange-Dirichlet theorem. Linearization and small oscillations; normal modes. Symmetry and first integrals: Noether theorem and Routh reduction. Introduction to the cariational principles of Mechanics: Euler-Lagrange equations, Hamilton principles; geodesics and constrained motions. Introduction to rigid body dynamics.

4. Introduction to Hamiltonian Mechanics: Legendre transformation. Hamilton equations. Poisson brackets. Conservation of volume.
Planned learning activities and teaching methods: Frontal lectures, with theory and exercises. Weekly suggestions on individual study, exercises, and self-assessment tests.
Additional notes about suggested reading: The program is covered in:
F. Fasso`, Dispense per il corso di Istituzioni di Fisica Matematica. CLEUP.

Reference material:
V.I. Arnold, Mathematical methods of Classical Mechanics (Springer).
G. Dell'Antonio, Elementi di Meccanica. I: Meccanica Classica (Liguori).
G. Gallavotti, Meccanica Elementare (Boringhieri).
A. Fasano, S. Marmi, B. Pelloni, Analytical Mechanics: An Introduction (Oxford Graduate Texts)
Textbooks (and optional supplementary readings)
  • F. Fasso`, Dispense per il corso di istituzioni di Fisica Matematica.. --: CLEUP, --. Cerca nel catalogo

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

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

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