ITALIANO

L'isegnamento verte su tre differenti argomenti della Fisica Matematica:
1) La teoria della stabilità del moto, come modello matematico si considera un sistema di equazioni differenziali del primo ordine non autonomo.
2) Elementi di meccanica analitica: si deducono le equazioni di Lagrange, le equazioni di Hamilton e si giustifica il principio di Hamiltom.
3) introduzione di alcune equazioni alle derivate parziali come modelli della meccanica del continuo.

W. Walter, Ordinary differential equations, Springer

H.Amann, Ordinary differential equations, an introduction to nonlinear analysis, de Gruyter Studies in Mathematics

F. John, Partial differential equations, Springer

L.C. Evans, Partial differential equations, GSM dell'Amer. Math. Society

M.E. Gurtin, An introduction to continuous mechanics, Academic Press

A.J. Chorin - J.E. Marsden, A mathematical introduction to fluid mechanics, Springer

A. Miranville - R. Temam, Modélisation mathématique et mécanique des milieux continus, Sprimger

A. Fasano - S. Marmi, Meccanica Analitica, Boricghieri.

Appunti del docente

Come naturale prosieguo delle nozioni apprese nei corsi di fisica matematica del triennio, uno degli obiettivi è approfondire alcuni metodi qualitativi per le equazioni differenziali ordinarie, considerate come modelli (dei più noti e consolidati) per la dinamica delle popolazioni o come modelli per alcuni fenomeni di evoluzione in meccanica classica. Lo studente al termine dell’apprendimento comprenderà che i modelli e le applicazioni possono essere differenti, ma i metodi matematici per la discussione dei modelli sono gli stessi.
Un secondo obiettivo è consentire l’apprendimento di ulteriori nozioni di meccanica analitica, in particolare si introduce lo studente al calcolo delle variazioni in meccanica classica. L’obiettivo è far apprendere la caratterizzazione tra le proprietà di minimo di opportuni funzionali e i moti naturali del problema dinamico.
Il terzo obiettivo è introdurre lo studente allo studio di alcune pde come modelli per la meccanica del continuo (diffusione del calore, meccanica dei fluidi). In questo modo si completa una preparazione di base della meccanica classica.
Tutti gli argomenti consentono allo studente di acquisire un bagaglio sufficiente per la comprensione matematica e la divulgazione matematica dei principali fenomeni presenti in natura.

Insegnamenti del triennio di laurea in Matematica

Lezione frontale

Prova orale con dissertazione su alcuni argomenti

1° argomento (3 crediti)
Introduzione alla nozione di modello
Costruzione di modelli in dinamica delle popolazioni
Richiami sulle equazioni differenziali
Dipendenza continua dai dati
Teoria della stabilità del moto alla Liapunov

2° argomento (3 crediti)
Equazioni di Lagrange come modello dei sistemi meccanici. Funzione di Lagrange.
Trasformata di Legendre, Hamiltoniana e equazioni di Hamilton
Il teorema di Liouville, Il teorema di Poincaré.
Introduzione ai principi variazionali
Equazioni di Eulero
Principio variazionale di Hamilton

3° argomento (2 crediti)
Trasformata di Fourier
Equazione del calore: soluzione fondamentale e proprietà di semigruppo
Equazioni di Navier-Stokes: formulazione variazionale per la stabilità in energia dei moti fluidi.

Italian

The teaching focuses on three different topics of Mathematical Physics:
1) The stability theory of solutions of non-autonomous systems of first-order differential equations, assumed as models of some natural phenomena, from the Newtonian Physics to the Ecologic models.
2) Elements of analytical mechanics: Lagrange's equations and Hamilton's equations are deduced, and the Hamilton principle is justified.
3) Introduction to the study of some partial differential equations considered as models of continuum mechanics.

W. Walter, Ordinary differential equations, Springer

H.Amann, Ordinary differential equations, an introduction to nonlinear analysis, de Gruyter Studies in Mathematics

F. John, Partial differential equations, Springer

L.C. Evans, Partial differential equations, GSM dell'Amer. Math. Society

M.E. Gurtin, An introduction to continuous mechanics, Academic Press

A.J. Chorin - J.E. Marsden, A mathematical introduction to fluid mechanics, Springer

A. Miranville - R. Temam, Modélisation mathématique et mécanique des milieux continus, Sprimger

A. Fasano - S. Marmi, Meccanica Analitica, Boricghieri.

Lecture notes

As a natural continuation of the notions learned in the three-year mathematical physics courses, one of the objectives is to delve into some qualitative methods for ordinary differential equations, considered as models (of the best known and consolidated) for population dynamics or as models for some evolution in classical mechanics. At the end of the learning, the student will understand that the models and applications may be different, but the mathematical methods for discussing the models are the same.
A second objective is to allow the learning of further notions of analytical mechanics, in particular the student is introduced to the calculus of variations in classical mechanics. The objective is to learn the characterization between the minimum properties of appropriate functionals and the natural motions of the dynamic problem.
The third objective is to introduce the student to the study of some PDEs as models for continuum mechanics (heat diffusion, fluid mechanics). In this way a basic preparation of classical mechanics is completed.
All topics allow the student to acquire sufficient knowledge for mathematical understanding and mathematical dissemination of the main phenomena present in nature.

As a natural continuation of the notions learned in the three-year mathematical physics courses, one of the objectives is to delve into some qualitative methods for ordinary differential equations, considered as models (of the best known and consolidated) for population dynamics or as models for some evolution phenomena related to the classical mechanics. At the end of the learning, the student will understand that the models and applications may be different, but the mathematical methods for discussing the models are the same.
A second objective is to allow the learning of further notions of analytical mechanics, in particular the student is introduced to the calculus of variations in classical mechanics. The objective is to learn the characterization between the minimum value of appropriate functionals and the natural motions of some dynamic problems.
The third objective is to introduce the student to the study of some PDEs as models for continuum mechanics (heat diffusion, fluid mechanics). In this way a basic preparation of classical mechanics is completed.
All topics allow the student to acquire sufficient knowledge for mathematical understanding and dissemination of the mathematical point of view concerning the main phenomena present in nature.

Basic mathematics knowledge usually acquired in university mathematics courses

classroom lecture

Oral examination on some topics

1st topic (3 credits)
Introduction to the notion of mathematical model.
Achieving of models in population dynamics
Review of differential equations
Continuous dependence on data
Liapunov theory of motion stability

2nd topic (3 credits)
Lagrange equations as a model of mechanical systems. Lagrange function.
Legendre transform, Hamiltonian and Hamilton equations
Liouville's theorem, Poincaré's theorem.
Introduction to some variational principles
Euler equations
Hamilton's variational principle

3rd topic (2 credits)
Fourier transform
Heat equation: fundamental solution and semigroup properties
Navier-Stokes equations: variational formulation for the energy stability of fluid motions.