The aim of the course is the illustration of the Lagrangian (and Hamiltonian, time permitting) formulation of the Newtonian mechanics. Qualitative analysis of the differential equations will be exploited, and fundamental questions like stability, variational principles and Kepler's problem will be considered.
Expected learning outcomes
Ability to study systems of differential equations, in particular dealing with their equilibria and the corresponding stability properties. Knowledge of machanics in its different formulations. Ability to stady constrained mechanical sytems by means of the lagrnagian formalism.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
1. Review of ordinary differential equations; existence, uniqueness and continuation of solutions. 2. Equilibria. Linearization, stability, Lyapounov theorem. 3. Review of Newtonian mechanics. Fundamental principles. Conservation laws. 4. Keplero (direct and inverse) problem. 5. Lagrangian formalism. Generalized coordinates. Lagrange equations. Symmetries and conservation laws; the case of Kepler problem. Equilibria and small oscillations; the case of the linear chain (and int connection with the vibrating string). 6. Introduction to the Hamiltonian formalism and variational principles.
Prerequisites for admission
Basic knowledge in analysis, geometry and linear algebra as provided in the first three semesters.
Lectures and problem classes. Students are strongly advised to attend the classes.
The final examination consists of two parts: a written exam and an oral exam.
- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in Lagrangian mechanics. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal. - The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate ideas, techniques, definitions and results presented during the course and will be possibly required to solve problems regarding Lagrangian mechnics in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.