00537 - Fundamentals of Mathematical Physics

Learning outcomes

At the end of the course the students master some important models of Mathematical Physics in the framework of classical Newtonian Mechanics (in the formalism of dynamical systems), of Analytical Mechanics and of Continuum Mechanics. Moreover they learn techniques and tools to solve the integrability problem of the mentioned systems.

Course contents

Elements of tensor analysis and differential geometry - Kinematics of discrete systems and continuum media: characterization of rigid motions - Newtonian mechanics in the formalism of dynamical systems -  Qualitative analysis of one dimensional motion and stability of equilibrium - Linear oscillator (dumped and forced) - The central force problem and the two body problem - Dynamics of rigid body with a fixed point - Euler's equations and their integrability - Analytical mechanics: the variational formalism from the Lagrangian and Hamiltonian point of view for holonomous systems with ideal constraints. Examples of integrable systems - The normal modes theorem and Liouville's theorem - Symplectic matrices - Canonical transformations and generating functions - Poisson brackets and their properties -  Introduction to the Hamilton-Jacobi theory.

Readings/Bibliography

Fasano-Marmi: Meccanica Analitica, Boringhieri
T. Ruggeri: Introduzione alla Termomeccanica dei continui, Monduzzi 2007
Lecture notes of the teacher

Teaching methods

The course consists of class-room lectures, where the basic elements of classical, analytical and continuum Mechanics are introduced. For each topic, after the theoretical presentation, several lectures are devoted to exercises, within the different formalisms, with the aim of making the students acquire the different methods of solution.

Assessment methods

The assessment method consists of a final written test of thee hours, where the use of didactical material is allowed, followed by an oral exam about the the main subjects covered within the course. The written test aims at ascertaining the abilities acquired in solving problems within the different topics. Students are admitted to the oral exam upon passing the written test. At the beginning of the course, the teacher will consider the opportunity of replacing the unique written test with two partial tests, each of three hours, during the academic year, related to the main formalisms of the Newtonian and Analytical Mechanics respectively.

Office hours

See the website of Franca Franchi