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ADVANCED MATHEMATICAL PHYSICS 1 (523SM)

A.A. 2019 / 2020

TAF* CFU ORE Periodo Professors Materiale didattico
ADVANCED MATHEMATICAL PHYSICS 1 - MOD. A (523SM-1) Caratterizzante 6 48 Secondo semestre Bertola Marco
ADVANCED MATHEMATICAL PHYSICS 1 - MOD. B (523SM-2) Caratterizzante 6 48 Secondo semestre Guzzetti Davide
Lingua insegnamento 

English

Obiettivi formativi 

Mod. A - The course aims to introduce the fundamental theory of partial differential equations (PDEs) of mathematical physics, discussing their classification and methods of solution.

Mod B -At the end of the course, the student should be able to solve problems of analytical mechanics (Lagrangian and Hamiltonian formalism, analytical and geometrical methods), symplectic geometry and Poisson geometry.

The student should be able to apply the acquired knowledge to the solution of exercises and relatively basic theoretical problems in the modern approach to mechanics and integrable systems.
The student will have acquired a sufficient autonomy to understand which techniques and theoretical background are required to face new problems in above mentioned areas.
The student will acquire the ability to organize a self contained exposition of the subjects he has studied, together with autonomy in finding, reading and understanding textbooks.

Prerequisiti 

Analysis 1, Linear Algebra.
First order ODEs and linear ODEs with constant coefficients. Differential geometry, differentiable manifolds, tensor calculus.
A background in Newtonian and Lagrangian mechanics is required, consisting in the program of the course in analytical mechanics of the Bachelor's Degree

Contenuti 

Mod. A
1. Fundamental linear equations; wave, Helmholts, heat, Laplace and Poisson equations. Classification of partial differential equations of second order.
2. Boundary value problems for second order PDEs. Cauchy problem and Cauchy–Kowalevskaya theorem. Boundary value problems for elliptic equations. Well posed problems.
3. Banach and Hilbert spaces. L2 and C spaces. Self–adjoint operators.
4. Separation of variables for Laplace, Poisson and wave equations.
5. Spectrum of Sturm–Liouville operators. Special cases and classical orthogonal polynomials. Spherical functions.
6. Topics in nonlinear differential equations. Burgers equation and Cole-Hopf transformation. Korteweg-de Vries equation.
7. (Time permitting). Solution of the Korteveg-de Vries equation: forward and inverse scattering. Riemann– Hilbert problems.

Mod. B
Lagrangian and Hamiltonian systems and their mathematical description in terms of differential geometry.

Metodi didattici 

Lectures on basic and more advanced theoretical topics will be complemented by exercises. Though there is plenty of textbooks, sometimes LaTex typed notes will be distributed. All the exercises will also be distributed, together with their detailed solutions.

Programma esteso 

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Modalità di verifica dell'apprendimento 

The exam consists of a written and oral part. The written component will assess the student’s proficiency in solving the exercises and applying the theory to a level comparable to the exercises worked out in class. The oral component will assess the knowledge of the theoretical aspects as well as the property of the expression of mathematical concepts.

Altre informazioni 

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Testi di riferimento 

Mod A:
Of the following texts we will follow primarily the first, with support material from the other two and possibly other material to be distributed during the course.
1. Boris Dubrovin’s course notes: http://people.sissa.it/∼dubrovin/fm1 web.pdf
2. A.Tikhonov, A.Samarskij, Equazioni della fisica matematica, Edizioni Mir, 1977.
3. R.Courant, D.Hilbert, Methods of Mathematical physics, New York Intersci. Publ. 1989.

Mod B:
B. Dubrovin: Dispense del corso di meccanica analitica (SISSA).
-- A. Fasano, S. Marmi: {Meccanica Analitica}.
-- L.D. Landau, E.M. Lifsits: {Meccanica}
-- W.M. Boothby: {An Introduction to Differentiable Manifolds and
Riemannian Geometry}.
-- V. Arnold: {Metodi Matematici della Meccanica Classica}.