Module page
Meccanica razionale
| academic year: | 2013/2014 |
| instructors: | Carlo Boldrighini, Piero Negrini |
| degree course: | Mathematics - DM 270/04 (triennale) |
| type of training activity: | caratterizzante |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/07 Fisica matematica |
| teaching language: | italiano |
| program: | I-Z |
| period: | II sem (04/03/2014 - 07/06/2014) |
Lecture meeting time and location
Presence: highly recommended
Module aims:
In Theoretical Mechanics course the student has the opportunity to use all the techniques he already learned to analyze relevant problems. The aim of these lectures is to introduce the student to the mathematical (qualitative and quantitative) methods of the Classical Mechanics . Newton equations are studied both for constrained and unconstrained systems. In particular we deal with some classical integrable problems, e.g. one-dimensional motions, rigid body problem in the case symmetry of the masses. The learner is also introduced to the variational calculus. Finally the stability of the stationary motions are analyzed from the linear and non linear point of view.
Module subject:
The Newtonian paradigma. Some recalls on the theory of first order differential systems.
Qualitative analysis of one-dimensional Newton equation: stationary solutions, their stability properties. Pictures of the orbits in the phase space. Linear systems and resonances.
The two body problem and the Kepler laws.
N-body systems. Cardinal equations.
Ideal constraints and the Lagrange equations.
Lagrangian systems and variational principles. Symmetries and first integrals.
The stability problem of equilibrium. Linear analysis and normal modes. Lagrange-Dirichlet theorem.
Rigid body Kinematics. Euler angles, lagrangian formulation and important cases of the dynamics of the rigid body.
Suggested reading:
P. Buttà, P. Negrini, Note del corso di Meccanica Razionale, Ed. Nuova Cultura
Type of course: standard
Examination tests:
- Testo del secondo esonero con svolgimento dello stesso.
- Testo della prova scritta del 23/06/2014 con svolgimento
- Testo della prova scritta del 8/07/2014 con svolgimento
- Testo della prova scritta del 9/09/2014 con svolgimento
- Testo della prova scritta del 23/09/2014 con svolgimento
- Testo della prova scritta del 10/11/2014 con svolgimento.
- Testo della prova scritta del 20/1/2015 con svolgimento.
Prerequisites: Calculus I, Linear Algebra, Mathematical Analysis I, Geometry I
Knowledge and understanding:
Successful students will be able to
build up models and then apply mathematical methods, both qualitative, quantitative and numerical.
Skills and attributes: Successful students will be able to: 1) study the pictures of the motions in the phase plane, to evaluate the periods of periodic solutions. 2) study the problem of the stability of the equilibrium by employing some methods of the Liapunov theory and by performing the spectral analysis to compute the frequencies of the oscillations around a stable equilibrium. 3) choose local coordinates for the configuration submanifolds of the constrained systems (e.g. Euler angles for SO(3), spherical coordinates, and so on) 4 4) recognize and use the variational nature of the Lagrange equations 5) exploit the existence of first integrals to reduce the Lagrange system.
Personal study: the percentage of personal study required by this course is the 65% of the total.