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ANALYTICAL MECHANICS

CODE 25911
ACADEMIC YEAR 2022/2023
CREDITS
  • 8 cfu during the 2nd year of 8760 MATEMATICA (L-35) - GENOVA
  • SCIENTIFIC DISCIPLINARY SECTOR MAT/07
    LANGUAGE Italian
    TEACHING LOCATION
  • GENOVA
  • SEMESTER 2° Semester
    PREREQUISITES
    Prerequisites
    You can take the exam for this unit if you passed the following exam(s):
    • PHYSICS 8758 (coorte 2021/2022)
    • MATHEMATICAL ANALYSIS 1 52474
    • GENERAL PHYSICS 1 72884
    • LINEAR ALGEBRA AND GEOMETRY 80275
    Prerequisites (for future units)
    This unit is a prerequisite for:
    • PHYSICS 8758 (coorte 2021/2022)
    • QUANTUM PHYSICS (A) 66560
    • ADVANCED CLASSIC PHYSICS 61739
    • QUANTUM PHYSICS 66559
    • QUANTUM PHYSICS (B) 66562
    TEACHING MATERIALS AULAWEB

    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

    Bibliographical informations will be given at the beginning of the lectures.

     

    TEACHERS AND EXAM BOARD

    Exam Board

    PIERRE OLIVIER MARTINETTI (President)

    SIMONE MURRO

    MARCO BENINI (President Substitute)

    LESSONS

    LESSONS START

    The class will start according to the academic calendar.

    Class schedule

    All class schedules are posted on the EasyAcademy portal.

    EXAMS

    EXAM DESCRIPTION

    The exam consists of two parts, writtern and oral. The aim of the first part is to assess the student's ability to use the tools presented in the lecture to independently solve mechanical problems. The aim of the second part is to assess the student's understanding of the concepts and results presented in the lecture, as well as the student's ability to reproduce the proofs of the main theorems.

    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.

    Exam schedule

    Date Time Location Type Notes
    10/01/2023 09:00 GENOVA Scritto riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    12/01/2023 09:00 GENOVA Orale riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    14/02/2023 09:00 GENOVA Scritto riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    16/02/2023 09:00 GENOVA Orale riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    20/06/2023 09:00 GENOVA Scritto
    22/06/2023 09:00 GENOVA Orale
    19/07/2023 09:00 GENOVA Scritto
    20/07/2023 09:00 GENOVA Orale
    12/09/2023 09:00 GENOVA Scritto
    13/09/2023 09:00 GENOVA Orale

    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.