CODE 107033 2024/2025 6 cfu anno 2 FISICA 8758 (L-30) - GENOVA MAT/07 GENOVA 2° Semester AULAWEB

## OVERVIEW

During these lectures, the student is introduced to the Lagrangian and Hamiltonian formulation of the classical mechanics.

Furthermore, the lectures contains also some elements of the theory of stability for dynamical systems, variational principles and Hamilton-Jacobi equation.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The teaching aims to make students acquire the ability to solve typical problems in classical physics by means of the tools provided by the formulation of Lagrangian and Hamiltonian mechanics

### AIMS AND LEARNING OUTCOMES

At the end of the learing path the student will be able to:

- describe the foundations of the Lagrangian and Hamiltonian formulation of classical mechanics

- describe the dynamics of classical systems by means of the Euler-Lagrange equation

- find the equilibrium configurations of Lagrangian systems

- analyze the stability of the equilibrium configurations of these systems

- formulate the equation of motion in the case of Hamiltonian mechanics

- know and use the canonical transforamtions

- know some advanced techniques to solve some motion equations like those furnished by the Hamilton-Jacobi equation

- characterize the studied equation of motion by means of some variational principles

### TEACHING METHODS

The course are organized in lectures given by the teachers where the theoretical part it will be presented and where its application to the resulutions of some exercises will be discussed.

### SYLLABUS/CONTENT

Introduction and some basic concepts

• Spacetime of the classical mechanics

Analytical mechanics of holonomic systems

• Holonomic systems and ideal constraints
• Euler-Lagrange equations
• Lagrange equation and balance equations
• Integrals of motion in the Lagrangian formalism

Introduction to stability of dynamical systems

• Equilibrium solution, critical points and their stability
• Small oscillations for a mechanical systems

Hamiltonian mechanics

• Legendre transformation and Hamilton's equations
• Poisson brackets
• Canonical transformations and generating functions
• Transformation law for the Hamiltonian
• Hamilton-Jacobi equation

Variational principles

• Lagrangian case and Hamiltonian case
• Canonical transformations and covariance of the action functional

The notes of the course will be made available within aul@web.

1) H. Goldstein, C. Poole, J. Safko, “Classical Mechanics”, 3rd edn. Addison-Wesley, San Francisco, (2002).

2) V. I. Arnold “Metodi Matematici della Meccanica Classica” Editori Riuniti University Press, (2010).

## LESSONS

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## EXAMS

### EXAM DESCRIPTION

The exam is usually formed by a written and by an oral part.

Students with learning disorders ("disturbi specifici di apprendimento", DSA) will be allowed to use specific modalities and supports that will be determined on a case-by-case basis in agreement with the Lecturers.

### ASSESSMENT METHODS

The written exams verifies the ability of the student to solve some exercises by means of the tecniques studied during the lectures.

The oral part is about the teoretical arguments presented during the lectures.