Mathematics

SERGIO SIMONELLA
SERGIO SIMONELLA

General targets: To acquire basic knowledge in classical mechanics.

Knowledge and understanding: Students who have passed the exam will be able to construct mathematical models not only for problems of mechanical nature, and to use analytic and qualitative methods of ordinary differential equations to deal with them.

Applying knowledge and understanding: Students who have passed the exam will be able: i) to perform the qualitative analysis on the phase space for one-dimensional conservative systems and to obtain quantitative estimates; ii) to study problems of stability of equilibrium points elementary methods of Liapunov; iii) to calculate frequencies of normal modes around stable equilibria; iv) to choose properly Lagrangian coordinates for particular configuration manifolds (like Euler angles for SO(3), spherical coordinates, etc.); iv) to recognize the variational nature of Lagrange equations and their implications; v) to use specific criteria for searching prime integrals in Lagrange equations and to perform the subsequent reduction to a smaller number of degrees of freedom.

Making judgements: Students who have passed the exam will have the basis to analyze the similarities between the topics covered in the course and the already acquired knowledges in analysis and geometry; they will also acquire important tools and ideas that have historically led to the solution of fundamental problems of classical mechanics.

Communication skills: Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of analytical mechanics.

Learning skills: The acquired knowledge will allow students who have passed the exam to face the study, at an individual level or in a master's degree course, of specialized aspects of classical mechanics and, more generally, of the theory of dynamical systems.

General objectives: To acquire basic knowledge in classical mechanics.

Knowledge and understanding: Students who have passed the exam will be able to formulate mathematical models of problems of a mechanical nature and to use analytical and qualitative methods of ordinary differential equations, as explained in the course.

Applying knowledge and understanding: Successful students will be able to: i) conduct qualitative in-phase analysis for one-dimensional conservative systems and make quantitative estimates; ii) study equilibrium stability problems with the (elementary) methods of Liapunov theory; iii) calculate frequencies of normal modes relative to stable equilibrium positions; iv) make an appropriate choice of Lagrangian coordinates in the case of particular configurational manifolds (in particular Euler angles for SO(3), spherical coordinates, etc.), recognize the variational nature of Lagrange equations and exploit the consequences deriving from it; v) use particular criteria in the search for first integrals of the Lagrange equations and operate the consequent reduction to a lower number of degrees of freedom.

Critical and judgment skills: Students who have passed the exam will have the basis for analyzing the analogies between the topics covered and the already acquired knowledge of analysis and geometry; they will also acquire important tools and ideas that have historically led to the solution of fundamental problems of classical mechanics.

Communication skills: Students who have passed the exam will have developed the ability to communicate concepts, ideas and methodologies of analytical mechanics.

Learning skills: The knowledge acquired will allow students who have passed the exam to study specialized aspects of classical mechanics and, more generally, the theory of dynamical systems, individually or in a master's degree course.

The course requires familiarity with the topics of the courses of calculus, mathematical analysis I and II, linear algebra and general physics I.

- Background on ordinary differential equations, equilibria and stability.
- Axiomatic formulation of Newtonian mechanics for systems of point particles.
- Cardinal equations and conservations laws.
- Qualitative analysis of systems with one degree of freedom.
- Central force motion and Kepler’s problem.
- Variational principles and Euler-Lagrange equations.
- Dynamics of constrained systems of point particles.
- Lagrangian systems, reduction to normal form, first integrals.
- Equilibria, stability and instability criteria, small oscillations.
- Symmetries and Noether’s theorem.
- Introduction to the dynamics of a rigid body.

P. Buttà, P. Negrini, Note del corso di meccanica razionale, Edizioni Nuova Cultura.
R. Esposito, Appunti dalle lezioni di meccanica razionale, Aracne Editrice.

V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer. L.D. Landau, E.M. Lifshitz, Mechanics. Vol. 1 (3rd ed.), Butterworth-Heinemann. G. Gallavotti, The Elements of Mechanics, Springer. E. Olivieri, Appunti di meccanica razionale, Aracne Editrice.

[E] R. Esposito, Appunti dalle lezioni di meccanica razionale, Aracne Editrice (disponibili in rete)
[G] G. Gallavotti, Meccanica Elementare. Editore Paolo Boringhieri, Torino.
[LL] L.D. Landau, E.M. Lifshitz, Fisica Teorica 1 - Meccanica, Editori Riuniti University Press.
[O] E. Olivieri, Appunti di meccanica razionale, Aracne Editrice.

The course is based on lectures in classroom, aiming to transfer to students the fundamental concepts of the discipline. Letture (60%), exercise sessions (40%).

Attendance at lessons is important for a good understanding of the course

The exam aims to evaluate learning through a written test (consisting in solving problems of the same type as those carried out in the exercises) and an oral test (consisting in the discussion of the most relevant topics illustrated in the course). The written test will last approximately two to three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place mid-course and the second immediately at the end of the course. The first intermediate test will focus mainly on the qualitative analysis of one-dimensional and central motions, the second on the Lagrangian systems with more degrees of freedom.
To pass the exam the student must demonstrate that he has acquired sufficient knowledge of the subjects and is able to perform at least the simplest of the assigned exercises.