#
Classical mechanics

A.Y. 2021/2022

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

Course syllabus and organization

### CORSO A

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**

Newton equations and energy for a mechanical system: Newton equations as a system of differential equations. Cauchy problem. Linear equation. Energy conservation in system with one degree of freedom. Energy conservation in systems with more degrees of freedom. Cardinal equations.

Lagrange equations. Deduction of Lagrange equations for (1) a particle in a general system of coordinates, (2) a system of particles subject to holonomic constraints. Jacobi integral. Conservation of energy. Kinetic energy as a positive definite quadratic form. Normal form of the Lagrange equations of a natural system. Equlibrium points of a lagrangian system and their stability. Linearization, small oscillations.

- Hamilton Equation: deduction of the equation; phase space. Flow of a vector field, Hamiltonian flow, Lie derivative. Poisson Breckets and their properties. Relationship between symmetries and constants of motion. Canonical transformations.

- Variational principles: Hamilton principle of least action for both Lagrange and Hamilton equation. 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 Galileo transformations. Some applications: bound on the speed of particles, addition of velocities, Lorentz contraction and time dilation. 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 equations. Deduction of Lagrange equations for (1) a particle in a general system of coordinates, (2) a system of particles subject to holonomic constraints. Jacobi integral. Conservation of energy. Kinetic energy as a positive definite quadratic form. Normal form of the Lagrange equations of a natural system. Equlibrium points of a lagrangian system and their stability. Linearization, small oscillations.

- Hamilton Equation: deduction of the equation; phase space. Flow of a vector field, Hamiltonian flow, Lie derivative. Poisson Breckets and their properties. Relationship between symmetries and constants of motion. Canonical transformations.

- Variational principles: Hamilton principle of least action for both Lagrange and Hamilton equation. 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 Galileo transformations. Some applications: bound on the speed of particles, addition of velocities, Lorentz contraction and time dilation. 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.

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. Eigenvectors and eigenvalues.

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. Eigenvectors and eigenvalues.

**Teaching methods**

Frontal lectures. There are also tutorials, in which some exemples are solved by the methods illustrated in the lectures.

**Teaching Resources**

Lecture notes by Bambusi available in 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 exam 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 in the lectures.

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 in the lectures.

MAT/07 - MATHEMATICAL PHYSICS - University credits: 7

Practicals: 24 hours

Lessons: 40 hours

Lessons: 40 hours

Professors:
Bambusi Dario Paolo, Montalto Riccardo

### CORSO B

Responsible

Lesson period

First semester

**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: 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.

- 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**

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, Gallone Matteo

Educational website(s)

Professor(s)

Reception:

Wednesday, 13.30-17.30

Room 1005, Department of Mathematics, Via Saldini 50, 20133, Milan