MODELS OF PHYSICAL MATHEMATICS
cod. 18975

Academic year 2024/25
3° year of course - First semester
Professor
Maria GROPPI
Academic discipline
Fisica matematica (MAT/07)
Field
Formazione modellistico-applicativa
Type of training activity
Characterising
72 hours
of face-to-face activities
9 credits
hub:
course unit
in ITALIAN

Learning objectives

Aim of the course is to describe the fundamental tools for the qualitative analysis of differential equations.
At the end, the student will be able to apply such tools to the formulation and the study of simple mathematical models.
The student will acquire the knowledge of foundations of Mathematical Physics, with a deep understanding of the basic applications of mathematical methods to the study of physical problems. Moreover, the student must become able to read and understand advanced text of Mathematical Physics.

Applying knowledge and understanding: the student must become able to produce formal proofs of results of Classical Mechanics and Mathematical Physics, and to expose, analyze and solve simple problems of Classical Mechanics with a clear mathematical formulation.
Moreover, the student will be able to apply computational tools to the mathematical modeling.

Making judgements: the student must become able to construct, develop and apply theoretical reasoning in the context of Mathematical Physics, with a deep ability to distinguish correct and wrong assumptions and methods.

Communication skills: the student must acquire the correct terminology and language of Mathematical Physics and the ability to expose their results and techniques to an audience, in both cases of qualified and unqualified audience.

Learning skills: the student must become able to autonomously continue the study of Mathematical Physics and in general to complete his preparation in Mathematics or in other scientific fields with an open minded approach, and must become able to gain knowledge from specialized text and journals.

Prerequisites

Basic calculus of the first year courses; mandatory propedeuticities: Mathematical Analysis 1, Rational Mechanics

Course unit content

Introduction to mathematical modelling through differential equations. The first part of the lectures concerns the Hamiltonian mechanics, the second the Liapunov’s stability theory for systems of ordinary differential equations, with applications to mathematical models in Mechanics, Population Dynamics and Epidemiology.

Full programme

Hamiltonian Mechanics:
Phase space, Legendre Transform, Hamiltonian function,
Canonical Hamiltonian System

Dynamical Systems. Equilibria and Stability. Lyapunov Methods.
Linear and nonlinear models in Mechanics.
Mathematical models in Population Dynamics.
Van der Pol equation.
Bifurcation theory, Hopf theorem, limit cycles.
Poincarè-Bendixon theorem.
Lorenz system and chaos.
Chaos indicators.

Discrete dynamical systems. Feigenbaum map.

Bibliography

Introduction to mathematical modelling through differential equations. The first part of the lectures is relevant to the Hamiltonian mechanics, Liapunov’s stability theory for systems of ordinary differential equations, with applications to mathematical models in Mechanics, Population Dynamics and Epidemiology.

Teaching methods

Lectures and exercises; laboratory of Matlab numerical simulation

Assessment methods and criteria

Oral exam and discussion of a project about a specific mathematical model in Applied Sciences.

Other information

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2030 agenda goals for sustainable development

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