Classical Mechanics
A.Y. 2022/2023
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 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
CORSO A
Responsible
Lesson period
First semester
More specific information on the delivery modes of training activities for academic year 2022/23 will be provided over the coming months, based on the evolution of the public health situation.
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. 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: 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 invariance of Maxwell's equations, and the lagrangian of a particle interacting with the electromagnetic field.
- 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. 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: 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 invariance of Maxwell's equations, and the lagrangian of a particle interacting with the electromagnetic field.
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) Elementari notions in vector algebra, in particular scalar product and vector product in ordinary space.
2) Notions from calculus, in particular the chain rule.
3) Elementari 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
Landau, Lifshitz "Meccanica", Editori Riuniti (or, english version, "Mechanics" Pergamon Press )
Carati, Galgani, "Appunti di Meccanica Analitica 1", downloadable from internet
Giorgilli "Appunti di Meccanica Analitica", from ariel
Carati, Galgani, "Appunti di Meccanica Analitica 1", downloadable from internet
Giorgilli "Appunti di Meccanica Analitica", from ariel
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 developped in the lectures to solving problems. The oral examination focus on the program topics, in order to ascertain the student understanding of the theory illustarted during the lectures.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 7
Practicals: 24 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Benedikter Niels Patriz, Paleari Simone
CORSO B
Responsible
Lesson period
First semester
The course will be delivered entirely remotely in case of restrictions due to Covid-19. The lectures will be offered in virtual classrooms (zoom platform) in synchronous connection.
Course syllabus
- 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. 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. Pseudometric and proper time. Lagrangian of the free particle, momentum, energy and rest energy. Four vector in space time: four velocity and four momentum.
Covectors in space-time. Lorentz transformation for covectors. Relativistic Doppler effect. Particle in a force field: relativistic invariant Lagrangian. Lorentz force. Four moment conservation: scattering.
- 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. Pseudometric and proper time. Lagrangian of the free particle, momentum, energy and rest energy. Four vector in space time: four velocity and four momentum.
Covectors in space-time. Lorentz transformation for covectors. 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) Elementari notions in vector algebra, in particular scalar product and vector product in ordinary space
2) Notions from calculus, in particular the chain rule.
3) Elementari notions in vector algebra, in particular scalar product and vector product in ordinary space
Teaching methods
Frontal lectures. There are also tutorials, in which some typical problems are solved by the methods illustrated in the lectures.
Teaching Resources
Landau, Lifshitz "Meccanica", Editori Riuniti (or, english version, "Mechanics" Pergamon Press )
Carati, Galgani, "Appunti di Meccanica Analitica 1", downloadable from internet
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 developped in the lectures to solving problems. The oral examination focus on the program topics, in order to ascertain the student understanding of the theory illustarted during the lectures.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 7
Practicals: 24 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Carati Andrea, Gubbiotti Giorgio
Professor(s)
Reception:
by appointment via e-mail
office 1024 (first floor, Via Cesare Saldini 50)
Reception:
Tuesday 2.30PM-4.30PM
Diparimento di Matematica "Federigo Enriques" Room 1040
Reception:
Contact me via email
Office 1039, 1st floor, Dipartimento di Matematica, Via Saldini, 50