Mathematical Physics 1
A.Y. 2023/2024
Learning objectives
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.
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.
lagrnagian formalism.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Elements of qualitative theory of solutions of ordinary differential equations: Linearization, stability, Lyapunof's theorem.
2. Newton's equations as differential equations: conservation of energy and its use for the qualitative study of solutions.
3. Kepler's problems and beyond.
3.1 Motion of a particle (planet) under the action of a central field of forces: study of the bonded orbits, proof of the fact that they are either periodic or dense in an annulus.
3.2 The particular case of the gravitational forces: proof of the fact that the Kepler's law are a consequence of Newton's equation.
4. Lagrangian formalism.
4.1 Generalized coordinates. Systems subject to Holonomic constraints. Deduction of Lagrange equations.
4.2 Qualitative properties of Lagrange equations: Jacobi's integral (conservation of energy), equilibria and their stability
4.3 Small oscillations
5. Elements of Hamiltonian formalism: equivalence between Hamilton and Lagrange equations, Poisson brackets, connection between symmetries and conserved qualities. The particular case of the momentum.
2. Newton's equations as differential equations: conservation of energy and its use for the qualitative study of solutions.
3. Kepler's problems and beyond.
3.1 Motion of a particle (planet) under the action of a central field of forces: study of the bonded orbits, proof of the fact that they are either periodic or dense in an annulus.
3.2 The particular case of the gravitational forces: proof of the fact that the Kepler's law are a consequence of Newton's equation.
4. Lagrangian formalism.
4.1 Generalized coordinates. Systems subject to Holonomic constraints. Deduction of Lagrange equations.
4.2 Qualitative properties of Lagrange equations: Jacobi's integral (conservation of energy), equilibria and their stability
4.3 Small oscillations
5. Elements of Hamiltonian formalism: equivalence between Hamilton and Lagrange equations, Poisson brackets, connection between symmetries and conserved qualities. The particular case of the momentum.
Prerequisites for admission
Basic knowledge in analysis, geometry and linear algebra as provided in the first three semesters.
Teaching methods
Lectures and problem classes.
Students are strongly advised to attend the classes.
Students are strongly advised to attend the classes.
Teaching Resources
Lecture notes available on the web page of the Ariel service.
Assessment methods and Criteria
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 dynamical systems, Lagrangian and Hamiltonian 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.
- 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 dynamical systems, Lagrangian and Hamiltonian 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.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Practicals: 24 hours
Lessons: 36 hours
Lessons: 36 hours
Professors:
Bambusi Dario Paolo, Montalto Riccardo
Educational website(s)
Professor(s)
Reception:
Wednesday, 13.30-17.30
Room 1005, Department of Mathematics, Via Saldini 50, 20133, Milan