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 different 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.

Dynamical Systems (DS) in Newtonian Mechanics, in Astrophysics, for magnetic field lines, and in ecology/bioastronomy, as the Lotka-Volterra model for two interacting species, prey-predator, towards SIR models in epidemiology.

Equilibrium configurations and linear/nonlinear stability analysis via the Ljapunov stability methods.

Topological classification of a singularity in R^2.

Qualitative analysis of one dimensional motions due to a positional force and stability of equilibrium.

The Period Theorem between two cosecutive inversion points.

Phase plane and equilibrium.

The simple pendulum.

Damped oscillations, forced oscillations. Resonance-

The central force problem and the two body problem as a reduction to the equivalent one-body problem.

Singular and non singular orbits in a central field.

Stability analysis of circular orbits  towards uniform circular motions.

Kepler's problem.

Apsidal angle towards closed/dense orbits.

Potentials admitting closed orbits: Theorem due to Bertrand.

Dynamics of rigid body with a fixed point - Euler's equations and their integrability.

Fom Newtonian Mechanics to Analytical Mechanics: the d'Alembert Principle towards the Lagrange Equations.

Equlibrium, stability and small oscillations.

Analytical mechanics: Lagrangian and Hamiltonian description.

The normal mode Theorem for the integrability of the linear system of l-oscillators: the special role of action-angle variables.

Legendre transformation: the Hamiltonian H.

Equivalence between the Lagrange Equations and the Hamilton Equations.

Liouville's Theorem and the role of the Hamilton Function for fixed constraints.

Poisson brackets, an important tool within the characterization of first integrals for the Hamiltonian flow and for their invariance under canonical transformations: Jacobi's Theorem.

The variational formalism from the Lagrangian and Hamiltonian point of view for holonomous systems with ideal constraints: Hamiltonian Actions A and S towards the Hamilton Principle.

The Jacobi metric.

Symplettic structure of the Hamiltonian phase space.

Canonical formalism: canonical transformations, the Lie condition and generating functions.

Hamilton-Jacobi theory and integrability.

The Hamilton-Jacobi equation: separation of variables and Jacobi's Theorem, namely integrability by quadrature.

Examples of integrable systems with one/two degree of freedom.

Readings/Bibliography

Fasano-Marmi: Meccanica Analitica, Boringhieri
T. Ruggeri: Introduzione alla Termomeccanica dei continui, Monduzzi 2007
Lecture notes (theory and exercises) of the teacher published on Virtuale.

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 two/three 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 final written test with two partial tests, during the academic year, related to the main formalisms of the Newtonian and Analytical Mechanics respectively.

Office hours

See the website of Franca Franchi