Skip to content Skip to navigation

ADVANCED MATHEMATICAL PHYSICS 1 (523SM)

A.Y. 2020 / 2021

TAF* Credits number Duration (in hours) Period Professors Teaching material
ADVANCED MATHEMATICAL PHYSICS 1 - mod. A (523SM-1) Programme specific subjects 6 48 Second semester Porta Marcello
ADVANCED MATHEMATICAL PHYSICS 1 - mod. B (523SM-2) Programme specific subjects 6 48 Second semester Guzzetti Davide
Teaching language 

English

Learning objectives 

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.

Prerequisites 

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

Contents 

Mod. A
1. Cauchy problem for systems of PDEs, Cauchy-Kovalevskaya theorem.
2. Linear differential operators.
3. Fundamental linear partial differential equations: Laplace, heat and wave equation.
4. The Fourier transform, applications to partial differential equations. The Schroedinger equation.
5. Topics in nonlinear partial differential equations. Burgers equation, Korteweg-de Vries equation

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

Teaching format 

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.

Extended Programme 

.

End-of-course test 

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.

Other information 

.

Texts/Books 

Mod A:
1. Boris Dubrovin's course notes: https://people.sissa.it/~dubrovin/fm1_web.pdf
2. L. C. Evans, Partial Differential Equations, AMS, 2010.
3. A. Tikhonov, A. Samarskij, Equazioni della fisica matematica, Edizioni Mir, 1977.
4. R. Courant, D. Hilbert, Methods of Mathematical Physics, New York Intersci. Publ., 1989.
5. E. H. Lieb, M. Loss, Analysis, AMS, 2001.

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}.