28375 - Mathematical Physics 2

Learning outcomes

A the end of the course the student will know the general methods of classical mechanics. He will know how to set up and solve any problem of motion with or without constraints.

Course contents

Summary of the kinematics of point masses and of the rigid body. Relative kinematics for a point and the Coriolis theorem. Fundamental principles of mechanics. Constraints and constraint forces. The postulate of the constraint forces. Method of the constraint forces to determine the conditions for the equilibrium. The principle of the virtual works. Equilibrium of systems under the action of conservative forces. Dynamics of point masses with or without constraits. Prime integrals. Motion of falling bodies. Motion of a point mass on a straight line under the action of a positional force: the Weierstrass analysis. Relative motion and elements of celestial mechanics. The two body problem and Kepler's laws. Dynamical characteristics of poit systems. The theorems of the linear momentum, of the angular momentum, of energy conservation. Koenig's theorem. The momentum of inertia and the motion of the rigid body. The Lagrange equations and the general set up for the motion of any mechanical system with or without constraints.

Readings/Bibliography

- Dario Graffi: Elementi di Meccanica Razionale (Patron);

- Dario Graffi: Esercizi di Meccanica Razionale (Patron).

The written lecture notes will be made availble for the download from the web.

The following link is recommended for the Weierstrass analysis:

http://progettomatematica.dm.unibo.it/sW/tesi.html

Teaching methods

The course consists of classroom lectures, where the basic elements of classical mechanics are introduced, up to the Lagrangian formalism included. More precisely, using the techniques of differential and integral calculus for functions of several real variables, on the basis of the fundamental space-time assumptions, Newton's law and D'Alembert-Lagrange principle, the main results are proved in the fields of kinematics, statics and dynamics of dynamical systems. For each topic first of all the theoretical aspects are illustrated and then several lectures are devoted to solving related excercises and specific problems, with the aim to emphasize the applied nature of the discipline and to acquire its method, that is the ability to express a practical problem in mathematical language and formalism to obtain its solution.

Assessment methods

The assessment method consists of a final written test of 3 hours (use of books, lecture notes, electronic devices not allowed) followed by an oral exam.
The written test aims at ascertaining the skills acquired in solving problems relative to the program of the course; students are admitted to the oral exam upon passing the written test.
The oral exam is aimed to assess the knowledge of the contents of the course, with specific attention to the theoretical aspects.
A further goal of both tests is to verify the acquisition of the general methods of classical mechanics and critical ability in solving the problems of motion. The final score is out of thirties and takes into account both performances of the written and oral parts.

Office hours

See the website of Emanuela Caliceti