Mathematics

DARIO BENEDETTO
DARIO BENEDETTO

General targets:
acquire basic specialist knowledge on
some classic topics of Physics-Mathematics.

Knowledge and understanding:
knowledge of the theory of compact self-adjoint operators, of the applications of this theory
to the theory of potential; basic knowledge of Hamiltonian Mechanics
and of Quantum Mechanics.

Applying knowledge and understanding:
the student will be able to
analyze the spectrum of operators, also for unbounbed operators; to determine
the eigenvalues of the Laplacian in domains with symmetries;
translate into Hamiltonian formalism
the Lagrangian problems and solve them for quadatrure;
discuss the solution of the
Schroedinger equation in simple but physically significant cases.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises, subject to written verification.

Making judgements:
ability to enucleate the most significant aspects of the potential theory
and of the theory of motion,
ability to reflect on similarities and differences between the classical case
and the quantum one.

Communication skills:
ability to enucleate the significant points of the theory,
to know how to illustrate the most interesting parts with appropriate examples,
to discuss mathematically the most subtle points.

Learning skills:
the acquired knowledge will allow the student to face
the mathematical-physcs courses on more specialized subjects,
and will allow the student to understand, even independently,
the physical relevance of mathematical questions discussed in other courses.

General objectives: acquire basic specialist knowledge
on some classic topics of Physics-Mathematics.

Specific objectives:

Knowledge and understanding of the foundations of Hamiltonian mechanics and kinetic theories.

Apply knowledge and understanding:
at the end of the course the student will be able to
translated into Hamiltonian formalism
some Lagrangian problems and portals to quadratures, and to use the main tools of Hamiltonian mechanics.
To develop these aspects, they are assigned and carried out during the course
appropriate exercises, subject to written verification.

Critical and judgment skills:
ability to highlight the significant aspects of the theory of motion.

Communication ability:
ability to highlight the significant points of the theory,
to be able to illustrate the most interesting parts with appropriate examples,
to discuss the finer points mathematically.

Learning ability:
the knowledge acquired will allow you to face
physics-mathematics courses on more specialized topics,
and will allow you to understand, even independently,
the physical relevance of mathematical questions
discuss in other courses.

Mathematical analysis: Fourier series. Lp spaces and Sobolev spaces.
Mathematicl physics: Basic knowledge on the equations of mathematical-physics, Green functions; harmonic functions; Lagrangian mechanics.

Details on http://brazil.mat.uniroma1.it/dario/#didattica

* First part (5 CFU)
The method of characteristics for first order PDEs; the Hamilton Jacobi equation; Hamilton equations;
the principle of stationary action.

Hamiltonian formalism: symplectic transformations, Poisson brackets, generating functions. Method of
Hamilton-Jacobi. Arnold-Liouville theorem, action-angle variables. Almost periodic motions on the torus.

Poincare's return theorems'.

* Second part (4 CFU)
Introduction to kinetic theories: the problem of the propagation of chaos.
Vlasov's equation and its validity.
The Boltzmann Equation.

Professor's notes, with exercises; see su http://brazil.mat.uniroma1.it/dario/#didattica

Courant, Hilbert: Methods of Mathematical Physics - Wiley
Komogorov, Fomin: Elementi di teoria delle funzioni e analisi funzionale - ERditori Riuniti
Salsa: Equazioni alle derivate parziali - Springer
Reed, Simon: Methods of modern mathematical physics, vol I, Functional Analisys - Elsevier
Kevin Cahill; Physical Mathematics, 2nd edition Cambridge University Press 2019
L. C. Evans Partial Differential Equation AMS Graduate Studies in Mathematics, vol. 19

Arnold: Meccanica Classica - Editori Riuniti
Landau: Meccanica - Editori Riuniti
Landau: Meccanica Quantistica - Editori Riuniti
Gallavotti: Meccanica Elementare - Boringhieri
Gantmacher: Meccanica Analitica - - Editori Riuniti

Lectures (80%), exercises (20%).

optional

The exam aims to evaluate the achievement of the objectives.

The written test, organized as a test, focuses on the topics of the exercises done in class.
The oral exam consists in the discussion of the most relevant topics,
both in the mathematical and in the physical aspects.
To achieve a score of 18/30, the student must demonstrate that he or she has acquired sufficient knowledge and skills on
basic topics of the course.
To achieve a score it seems to 25/30, the student must demonstrate to have acquired a good knowledge and competence on
basic topics of the course.
To achieve a score of 30/30 cum laude, the student must demonstrate that he has acquired excellent competence and knowledge of all the topics covered during the course.

see the teacher's web pages and classroom of the course