
Course unit
FUNDAMENTS OF MATHEMATICAL PHYSICS
SCP4065477, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2018/19
ECTS: details
Type 
ScientificDisciplinary 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 
Hours of Individual study 
Shifts 
Lecture 
9.0 
72 
153.0 
No turn 
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)
BERNARDI
OLGA
(Supplente)
CARDIN
FRANCO
(Supplente)
FAVRETTI
MARCO
(Supplente)
GUZZO
MASSIMILIANO
(Supplente)
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. Velocitydependent potentials: electromagnetic forces in the Lagrangian formalism. Equilibria and stability: LagrangeDirichlet theorem. Linearization and small oscillations; normal modes. Symmetry and first integrals: Noether theorem and Routh reduction. Introduction to the cariational principles of Mechanics: EulerLagrange 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 selfassessment 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, .

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
Sustainable Development Goals (SDGs)

