Mathematics

EMANUELE CAGLIOTI

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: Acquire basic knowledge in classical mechanics.

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

Apply knowledge and understanding: Students who pass the exam will be able to: i) conduct qualitative analysis in the phase plan for conservative one-dimensional 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 varieties (in particular Euler angles for SO(3), spherical coordinates, etc.), recognize the variational nature of Lagrange's equations and exploit the consequences that derive from it; v) use particular criteria in the search for first integrals of Lagrange's equations and carry out 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 to analyze the analogies between the topics covered and the knowledge already acquired 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 ability: The knowledge acquired will allow students who have passed the exam to study, individually or in a master's degree course, specialized aspects of classical mechanics and, more generally, the theory of dynamic systems.

Useful prerequisites are the first two courses of Mathematical Analysis and Physics.

- Basic notions on ordinary differential equations, equilibria and stability;
- Simple differential equations;
- Axiomatic formulation of Newtonian mechanics for systems of material points;
- Qualitative analysis of one-dimensional motions;
- Central motions and Kepler's problem;
- Variational principles and Euler-Lagrange equations;
- Dynamics of constrained systems;
- Lagrangian systems;
- Equilibrium, stability and instability, small oscillations;
- Euler equations for the rigid body;
- Hamiltonian systems: basic notions.

P. Buttà, P. Negrini, Note del corso di meccanica razionale, Edizioni Nuova Cultura.

V.I. Arnold, Mathematical methods of classical mechanics, Springer.
H. Goldstein, Classical Mechanics, Pearson.

Lectures on the theoretical concepts with classroom exercises.

Attendance at lessons is strongly recommended for a good understanding of the teaching contents.

The exam is an oral exam and it consists of the discussion of some of the main topics presented in the course.
To pass the exam the student has to achieve a grade no smaller than 18/30.
The student has to show to have acquired a sufficient knowledge of the arguments presented in the course and to be able to apply the methods learned in the course to the most simple examples considered in it.
To get a grade of 30/30 cum laude, the student has to show an excellent knowledge of the topics of the course and to be able to explain these topics in a coherent way.
Information above can change due to Covid.

Esempio di domande all'orale: Ricavare la prima legge di Keplero - Dimostrare il Teorema di Noether - Ricavare le equazioni di Eulero per un corpo rigido
Esempi di esercizi: risolvere un problema meccanico scrivendo la Lagrangiana, le equazioni del moto, trovando i punti di equilibrio e discutendone la stabilità, svolgendo l'analisi qualitativa, e trovando traiettorie particolari.