Classical Mechanics
A.Y. 2024/2025
Learning objectives
To use mathematical methods for the study of phisical
problems. Furthermore to learn the basic facts about the theory of
Relativity and the tools needed in order to begin the study of Quantum
Mechanics.
problems. Furthermore to learn the basic facts about the theory of
Relativity and the tools needed in order to begin the study of Quantum
Mechanics.
Expected learning outcomes
To be able to use mathematical methods for the study of phisical
problems. To be able to study the dynmics of simple mechanical
systems. To have a basic knowledge of special relativity. To know the
tools needed in order to begin the study of Quantum Mechanics
problems. To be able to study the dynmics of simple mechanical
systems. To have a basic knowledge of special relativity. To know the
tools needed in order to begin the study of Quantum Mechanics
Lesson period: First 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
CORSO A
Responsible
Lesson period
First semester
Course syllabus
- Introduction to dynamical systems: equilibria and stability. Classification for linear systems. Phase portraits for systems with one degree of freedom.
- Lagrange equation: deduction starting from Newton equation, in the case of a point on a smooth surface; deduction in the general case of holonomic constraint. Jacobi energy. The case of keplerian potential: bounded and scattering motions, scattering cross section. Symmetries and reductions. Equilibrium points and normal modes of oscillations.
- Hamilton Equation: deduction of the equation; phase space and Liouville theorem; Poisson brackets and first integrals; canonical transformation. Relation between simmetries and conserved quantities.
- Variational principles: Hamilton principle of least action for both Lagrange and Hamilton equation. Application to the canonical transformation (Lie theorem and generating functions). Principle of Mapertius. Hamilton principle for the vibrating string.
- Relativity: recalling Lorentz transformations (space-time, inertial systems and the the principle of invariant light speed; deduction of the Lorentz transformations and comparison with the Galileo ones; some applications: bound on the speed of particles, addition of velocities, Lorentz contraction and time dilatation). Geometrical interpretation in space-time: pseudometric and proper time. Twin paradox. Lagrangian of the free particle, momentum, energy and rest energy. Four velocity and four momentum. Relativistic Doppler effect. Particle in a force field: relativistic invariant Lagrangian. Lorentz force. Four moment conservation: scattering.
- Lagrange equation: deduction starting from Newton equation, in the case of a point on a smooth surface; deduction in the general case of holonomic constraint. Jacobi energy. The case of keplerian potential: bounded and scattering motions, scattering cross section. Symmetries and reductions. Equilibrium points and normal modes of oscillations.
- Hamilton Equation: deduction of the equation; phase space and Liouville theorem; Poisson brackets and first integrals; canonical transformation. Relation between simmetries and conserved quantities.
- Variational principles: Hamilton principle of least action for both Lagrange and Hamilton equation. Application to the canonical transformation (Lie theorem and generating functions). Principle of Mapertius. Hamilton principle for the vibrating string.
- Relativity: recalling Lorentz transformations (space-time, inertial systems and the the principle of invariant light speed; deduction of the Lorentz transformations and comparison with the Galileo ones; some applications: bound on the speed of particles, addition of velocities, Lorentz contraction and time dilatation). Geometrical interpretation in space-time: pseudometric and proper time. Twin paradox. Lagrangian of the free particle, momentum, energy and rest energy. Four velocity and four momentum. Relativistic Doppler effect. Particle in a force field: relativistic invariant Lagrangian. Lorentz force. Four moment conservation: scattering.
Prerequisites for admission
1) Elementary notions on Newton's equations, momentum, angular momentum, kinetics and potential energy for a system of points. In particular, the potential energy for the two-body internal forces.
2) Notions from calculus, in particular the chain rule.
3) Elementary notions in vector algebra, in particular scalar product and vector product in ordinary space.
2) Notions from calculus, in particular the chain rule.
3) Elementary notions in vector algebra, in particular scalar product and vector product in ordinary space.
Teaching methods
Lectures. There are also tutorials, in which some typical problems are solved by the methods illustrated in the lectures.
Teaching Resources
Giorgilli "Appunti di Meccanica Analitica", from ariel
Landau, Lifshitz "Meccanica", Editori Riuniti (or, english version, "Mechanics" Pergamon Press )
Carati, Galgani, "Appunti di Meccanica Analitica 1", downloadable from internet
Landau, Lifshitz "Meccanica", Editori Riuniti (or, english version, "Mechanics" Pergamon Press )
Carati, Galgani, "Appunti di Meccanica Analitica 1", downloadable from internet
Assessment methods and Criteria
The examination consists in a written and an oral test. The written test consist in solving some exercises, in order to ascertain the ability of the student to apply the methods developed in the lectures to solve problems. The oral examination focus on the program topics, in order to ascertain the student understanding of the theory illustrated during the lectures.
Access to the oral part is subject to the achievement of a favorable outcome in the written section. To pass the examination, both the written and oral components must be successfully completed. The final marks will be assigned on a numerical scale of 0-30 and will be communicated immediately following the oral examination.
Access to the oral part is subject to the achievement of a favorable outcome in the written section. To pass the examination, both the written and oral components must be successfully completed. The final marks will be assigned on a numerical scale of 0-30 and will be communicated immediately following the oral examination.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 7
Practicals: 24 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Lill Sascha, Paleari Simone
CORSO B
Responsible
Lesson period
First semester
MAT/07 - MATHEMATICAL PHYSICS - University credits: 7
Practicals: 24 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Carati Andrea, Gubbiotti Giorgio
Professor(s)
Reception:
Tuesday 2.30PM-4.30PM
Diparimento di Matematica "Federigo Enriques" Room 1026
Reception:
Contact me via email
Office 1039, 1st floor, Dipartimento di Matematica, Via Saldini, 50