Mathematical Physics 1
A.Y. 2024/2025
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 can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Mechanics as a mathematical discipline, from ancient Greeks to Newton and Lagrange: modern mathematical form of the principles of dynamics, qualitative study of 1-dimensional motions, planar dynamic systems.
2. Rigid body motion: fundamental formula for rigid bodies, Euler angles, and inertia ellipsoids.
3. Lagrangian mechanics: derivation of Euler-Lagrange equations from the principle of virtual work, conservation of energy for natural Lagrangians, cyclic coordinates, reduced Lagrangian.
4. The two-body problem: derivation from spatial symmetries, reduction to a 1-dimensional problem, superintegrability of Kepler-Coulomb's problem and three laws of Kepler, Rutherford scattering.
5. Variational formulation of Lagrangian mechanics.
6. Hamiltonian mechanics: Legendre transformation, Hamilton equations, Poisson brackets, canonical transformations.
7. Stability for mechanical systems: Dirichlet theorem, small oscillations and normal modes.
2. Rigid body motion: fundamental formula for rigid bodies, Euler angles, and inertia ellipsoids.
3. Lagrangian mechanics: derivation of Euler-Lagrange equations from the principle of virtual work, conservation of energy for natural Lagrangians, cyclic coordinates, reduced Lagrangian.
4. The two-body problem: derivation from spatial symmetries, reduction to a 1-dimensional problem, superintegrability of Kepler-Coulomb's problem and three laws of Kepler, Rutherford scattering.
5. Variational formulation of Lagrangian mechanics.
6. Hamiltonian mechanics: Legendre transformation, Hamilton equations, Poisson brackets, canonical transformations.
7. Stability for mechanical systems: Dirichlet theorem, small oscillations and normal modes.
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 encouraged to attend the classes.
Teaching Resources
- G. Grioli Lezioni di Meccanica Razionale, Raffaello Cortina editore (rigid bodies and Euler angles, not available in English, please refer to Idorov's book);
- E. Goldstein, C. Safko, J. Poole, Classical Mechanics, Addison Wesley (Lagrangian and Hamiltonian mechanics);
- F. R. Gantmacher, Lectures in Analytical Mechanics, Mir Editions (stability).
General references:
- I. E. Idorov, Fundamental Laws of Mechanics
- L. D. Landau e E. M. Lifshitz, Theoretical Physics Vol. 1 - Mechanics, Pergamon Press;
- V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer.
- E. Goldstein, C. Safko, J. Poole, Classical Mechanics, Addison Wesley (Lagrangian and Hamiltonian mechanics);
- F. R. Gantmacher, Lectures in Analytical Mechanics, Mir Editions (stability).
General references:
- I. E. Idorov, Fundamental Laws of Mechanics
- L. D. Landau e E. M. Lifshitz, Theoretical Physics Vol. 1 - Mechanics, Pergamon Press;
- V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer.
Assessment methods and Criteria
The exam consists of a written test and an oral examination.
- In the written test, some open-ended and/or closed-answer exercises will be assigned to verify the ability to solve problems in Lagrangian and Hamiltonian mechanics. The duration of the written test is proportional to the number and structure of the exercises set, but will not exceed three hours. The results of the written tests will be communicated on SIFA through the UNIMIA portal. The written test will have no marks in thirty-sixths, but only a pass/fail assessment.
- Only students who have passed the written test (or intermediate tests) from the same examination session will proceed to the oral examination. Students admitted to the oral must take the test within 12 months. During the oral examination, you will be asked to explain ideas, techniques, definitions and results of the teaching content, and may also be required to solve some problems in Lagrangian mechanics in order to assess your knowledge and understanding of the topics covered, as well as your ability to apply them.
The exam is considered passed if both the written test and the oral examination are successfully completed. The mark will be expressed in thirty-sixths and communicated immediately after the oral examination.
- In the written test, some open-ended and/or closed-answer exercises will be assigned to verify the ability to solve problems in Lagrangian and Hamiltonian mechanics. The duration of the written test is proportional to the number and structure of the exercises set, but will not exceed three hours. The results of the written tests will be communicated on SIFA through the UNIMIA portal. The written test will have no marks in thirty-sixths, but only a pass/fail assessment.
- Only students who have passed the written test (or intermediate tests) from the same examination session will proceed to the oral examination. Students admitted to the oral must take the test within 12 months. During the oral examination, you will be asked to explain ideas, techniques, definitions and results of the teaching content, and may also be required to solve some problems in Lagrangian mechanics in order to assess your knowledge and understanding of the topics covered, as well as your ability to apply them.
The exam is considered passed if both the written test and the oral examination are successfully completed. The mark will be expressed in thirty-sixths and communicated immediately after the oral examination.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Practicals: 24 hours
Lessons: 36 hours
Lessons: 36 hours
Professors:
Gubbiotti Giorgio, Latini Danilo
Professor(s)
Reception:
Tuesday 2.30PM-4.30PM
Diparimento di Matematica "Federigo Enriques" Room 1040