OVERVIEW

In this lecture, we will study the mathematical structures emerging from classical (i.e. newtonnian) mechanics. They offer a sistematic way to confront difficult problems in physics, like constrained movements. We will  study in particular the approches developped by Lagrange, then Hamilton.

There will be an introduction to the theory of stability for some classical dynamical sistems. A part of the course is devoted to the calculus of variations: this is the key to some pure mathematical problems (such as the search of extremal quantities) and the gate to some fundamental theorems in mathematical physics, like Noether theorem.

AIMS AND CONTENT

LEARNING OUTCOMES

In this teaching will be dealt with the foundations of both Lagrangian and Hamiltonian analytical mechanics and the theory of stability.

AIMS AND LEARNING OUTCOMES

The aim of the lecture is to present analytical mechanics, both from the Lagrange and the Hamiltonian point of view, and its applications to the solution of mechanical problems.
Starting from Newton's laws and the analysis of constrained systems, the Lagrangian formalism is introduced along with the Euler-Lagrange equations. The existence of (local) solutions for Euler-Lagrange equations is examined in detail, focusing in particular on the structure of the kinetic energy. During the course several physically interesting examples are discussed, both concerning systems of point particles and rigid bodies. Furthermore, stability theory à la Ljapunov for autonomous dynamical systems is developed, including its application to small oscillations around stable equilibrium configurations of a mechanical system.
The transition from the Lagrangian to the Hamiltonian formalism is achieved through the Legendre transform. This allows us to deduce Hamilton equations from Euler-Lagrange equations. The natural symplectic structure that appears manifestly in Hamiltonian mechanics is formalized introducing Poisson brackets, whose properties are analyzed in detail. Therefore, we are naturally led to introduce canonical transformations as those coordinate transformations that preserve the form of the Poisson brackets. Further equivalent characterizations of the class of canonical transformations are derived, including the one based on generating functions. The latter characterization of canonical transformations brings us to Hamilton-Jacobi equations, whose goal is to single out a system of canonical coordinates such that Hamilton equations become trivial.
The lecture will also introduce variational principles, both in the Lagrangian and in the Hamiltonian formalism, with some applications to problems both of geometric and of physical flavor, culminating in the demonstration of Noether theorem.
Throughout the lecture, weekly exercise classes are offered, whose aim is to prepare the student to independently solve a wide range of problems of mechanical nature using the techniques presented during the lecture.
 

TEACHING METHODS

The lecture consists of taught classes. Part of the classes are of theoretic nature (approximately 48 hours). The purpose of those is to present the theoretic aspects of analytical mechanics, along with some concrete examples. This theoretic part is complemented by weekly exercise classes (approximately 24 hours), whose purpose is to show how to solve concrete analytical mechanics problems using the tools presented in the theoretic part of the lecture.

SYLLABUS/CONTENT

Mechanics of the massive point

• Spacetime of classical physics
• Cinematics
• Dynamics
• Relative mechanics 
• Energy theorems

Analytical mechanics of holonomous systems

• Motion of the constrained point 
• Lagrangian mechanics
• Relative lagrangian mechanics
• Constants of motion

Variational principle

• Lagrangian case
• Noether theorem

Introduction to equilibrium

• Equilibrium state
• Stability of critical points
• Small oscillations

Hamiltonian mechanics

• Hamilton equations
• Canonical transformation
• Hamilton-Jacobi theory

RECOMMENDED READING/BIBLIOGRAPHY

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

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

V. Moretti “Meccanica analitca ”, Springer , (2020). 

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of two parts, writtern and oral.

The written part lasts 3h. Manuscritps (or printed) notes of the lectures are allowed, nut no electronic device.

Sucessfully passing the written exam gives access to the oral exam (at any posterior session, not necessarily the one following the written session)

The later consist in two questions (one by teachers).

ASSESSMENT METHODS

The assessment is made of two stages. The first stage consists of a written test, during which the student's task is to solve mechanical problems using the tools presented in the lecture. The second stage consists of an oral test, during which the student's task is to demonstrate her/his knowledge of the subject, as well as her/his ability to reproduce the proofs of the main results presented throughout the lecture.

FURTHER INFORMATION

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.