CODE  25911 

ACADEMIC YEAR  2021/2022 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/07 
LANGUAGE  Italian 
TEACHING LOCATION 

SEMESTER  2° Semester 
PREREQUISITES 
Prerequisites
You can take the exam for this unit if you passed the following exam(s):
Prerequisites (for future units)
This unit is a prerequisite for:

TEACHING MATERIALS  AULAWEB 
In this course will be dealt with the foundations of both Lagrangian and Hamiltonian analytical mechanics and the theory of stability.
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 EulerLagrange equations. The existence of (local) solutions for EulerLagrange 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 EulerLagrange 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 HamiltonJacobi equations, whose goal is to single out a system of canonical coordinates such that Hamilton equations become trivial.
The last part of the lecture focuses on variational principles, both in the Lagrangian and in the Hamiltonian formalism, with some applications to problems both of geometric and of physical flavor.
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.
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.
Introduction and basic concepts
Analytical mechanics of holonomic systems
Introduction to stability of dynamical systems
Hamiltonian mechanics
Variational principles
Office hours: On appointment
PIERRE OLIVIER MARTINETTI (President)
SIMONE MURRO
MARCO BENINI (President Substitute)
The class will start according to the academic calendar.
All class schedules are posted on the EasyAcademy portal.
The exam consists of two parts. 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.
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.
Date  Time  Location  Type  Notes 

08/06/2022  09:00  GENOVA  Scritto per Fisica con accettazione online  
14/06/2022  09:00  GENOVA  Scritto per Matematica con accettazione online  
17/06/2022  09:00  GENOVA  Orale per Matematica  
12/07/2022  09:00  GENOVA  Scritto per Fisica con accettazione online  
20/07/2022  09:00  GENOVA  Scritto per Matematica con accettazione online  
21/07/2022  09:00  GENOVA  Orale per Matematica  
06/09/2022  09:00  GENOVA  Scritto per Matematica con accettazione online  
07/09/2022  09:00  GENOVA  Scritto per Fisica con accettazione online  
07/09/2022  09:00  GENOVA  Orale per Matematica 