OVERVIEW

Advanced mathematical methods of physics (code 61843) has credit value 6 and it is taught in the first semester of the first or second  year of the LM.

Lectures are usually given in Italian. Should it be requested, they can be given in english.

AIMS AND CONTENT

LEARNING OUTCOMES

The calculus of variations is a general method to derive differential and partial differential equations 

used in physics.

We will show how to solve these equations using the theory of distributions and the Green's function method.

AIMS AND LEARNING OUTCOMES

The course is divided in two parts:

The first part  is on the applications of calculus of variations to mechanics and classical field theory.

The second one is on the introduction  of the distributions and the Green's functions.

PREREQUISITES

Mathematical methods of Physics.

TEACHING METHODS

Teaching is done in the traditional (chalk and blackboard) way. Students are (very) strongly suggested to attend the class.

SYLLABUS/CONTENT

Functionals. Variation of a functional. Simple examples.

Euler-Lagrange equations.

Symmetries and Noether's theorem.

The vibrating string.

Klein-Gordon Lagrangian. Retarded, advanced and Feynmann Green's functions.

Introduction to the theory of distributions.

Examples:  Dirac's Delta function and its derivatives.

Green's formulae.

Dirichlet and Neumann problem.

The wave equation.

Vibrations of a circular membrane.

Gross-Pitaevskii Lagrangian. Gross-Pitaevskii equations.

U(1) symmetry and conserved current.

RECOMMENDED READING/BIBLIOGRAPHY

Suggested textbooks

Elsgolc, Calculus of variations.

Butkov, Mathematical physics.

TEACHERS AND EXAM BOARD

Exam Board

NICODEMO MAGNOLI (President)

ANDREA AMORETTI

PIERANTONIO ZANGHI' (President Substitute)

LESSONS

LESSONS START

First semester, usually last week of September

Class schedule

ADVANCED MATHEMATICAL METHODS IN PHYSICS

EXAMS

EXAM DESCRIPTION

Written examination.Optional oral examination.

ASSESSMENT METHODS

Methodology of ranking

The explicit aim of the course is to  carry the students to the level of making calculations and solving problems. This is why the essential part of the exam is the written one, where  the student is asked  to make calculations and  to solve explicitly problems.

The text is divided into three levels, each level corresponds to a range of marks. (These ranges of marks are clearly indicated in the text given to each student).

The oral part is typically a small correction of the written examination and it is not mandatory.

Exam schedule