Physics

ANDREA CRISANTI

GENERAL OBJECTIVES:
The main objective of Advanced Mathematical Methods for Physics is that of providing an introduction to up-to-date computational methods that are used in research areas of current interest. Three different courses are offered.

The goal of the third course is to provide the students with the theoretical background of perturbative and asymptotic analysis used in many fields of theoretical physics:
a) Definition and properties of the perturbative and asymptotic exapansions used in theoretical physics;
b) Introduction to some asymptiotic methods --- Boundary Layers, WKB, Multiple Scale, Renormalization Group --- and analisys of their filds of applicability.
SPECIFIC OBJECTIVES:
A - Knowledge and understanding
OF 1) To know and understand the foundations of the perturbative methods
OF 2) To know and understand the foundations asymptotic analysis
OF 3) To know and understand the Boundary Layer Theory
OF 4) To know and understand the WKB method
OF 5) To know and understand the Multiple Scale method
OF 6) To know and understand the Renormalization Group and it connections with asymptotica analysis.

B - Application skills
OF 7) Application of the asymptotic analysis to the solution of comlex problems
OF 8) Application of the Boundary Layer Theory, WKB method and Multiple Scale method to the study of simple problems.
OF 9) Application of the Renormalization Group method to the asymptotic analysis of the solution of simple ordinary differential equations.

C - Autonomy of judgment
OF 10) Ability to analize a simple perturbative problem.
OF 11) Ability to evaluate the structure of a simple perturbative problem and use the more appropriate method to its study.
D - Communication skills
OF 12) Ability to create an effective presentation of the results of a theoretical project
OF 13) Ability to present the basis of the asymptotic analysis and some of its methods.
E - Ability to learn
OF 14) Autonomous improvement in the study of perturbation method
OF 15) Autonomous improvement in the use of asymptotic analysis in more comlex problems
OF 16) Autonomous improvement in reading and understanding research articles and reviews

Knowledge of the basic analytical and numerical calculus from courses of the Laurea in Physics

Perturbative expansions
Asymptotic expansions
Boundary Layer problems
WKB method
Multiple Scale analysis
Renormalization Group and asymptotic expansions

Advanced Mathematical Methods For Scientists and Engineers
C. Bender and S.A. Orszag
(Springer, 1999)

Perturbation Methods
Ali. H. Nayfeh
(Wiley, 2000)

Lectures and discussion of exercises in classroom.

Attendance to the lectures is not mandatory but strongly recommended.

The examination consists of an interview on the most relevant topics
presented in the course. To pass the exam, the student must be able to
present arguments and repeat calculations discussed and explained
during the course. The student will be asked to apply the
methods learned during the course to exercises or to examples and
situations similar to those that were discussed in the course.
The evaluation takes into account:
- Correctness and completeness of the concepts discussed by the
student;
- clarity and rigor of presentation;
- analytical development of the theory;
- problem-solving skills (method and results).