The aim of the course is to present the basic mathematical models of classical mechanics, in their theoretical aspects and in their applications.The expected learning outcomes are as follows: 1. knowledge of the Lagrangian formulation of the motion of a system of points; 2. knowledge of the study of equilibrium and relative stability; 3. acquisition of the basics of the Hamiltonian formalism of mechanics.
Course Prerequisites
Analysis 1, Analysis 2, Linear Algebra.
Teaching Methods
The teaching uses theoretical lessons on the blackboard and exercises preferably on exam topics. The details of the topics covered are indicated on the Kiro platform, where the texts of the exercises proposed as exercises to be discussed later in class are also uploaded. Attendance at lessons and exercises is strongly recommended and active participation by students is constantly encouraged.
Assessment Methods
Learning is tested through a written test and an oral test. The written test consists of four problems, formulated to test preparation on the basic topics of the course. Specifically, an exercise on the Lagrangian formalism, an exercise on equilibrium and stability, an exercise on the analysis of one-dimensional motions or central motions, an exercise on Hamiltonian formalism. The duration of the written test is two and a half hours. The oral test is designed to test theoretical knowledge of the topics covered, expository ability and knowledge of the language and specific techniques covered in the course. Exercises from previous exam topics are proposed and solved in detail during the course. The oral test questions are chosen from the topics listed in the course syllabus.
Texts
1.Fasano A., Marmi S.,: "Meccanica Analitica", Bollati Boringhieri. 2.Goldstein H., Poole C., Safko J.: "Meccanica Classica", Zanichelli. 3.Gantmacher F.R.: "Lezioni di Meccanica Analitica", Editori Riuniti. 4.Lanczos C., : "The variational principles of Mechanics, Dover.
Contents
Kinematics of a point. Dynamics: fundamental principles. The motion of a free particle. Constraints. Multi particles systems. Rigid systems. Cardinal equations of dynamics. Lagrange's equations. Some classical problems: the problem of two bodies. Equilibrium and stability. Hamilton's principle. Hamilton's equations. Canonical tranformations. Poisson brackets.
Extended summary
Kinematics of a point. Frenet's frame. Constraints and their classification. The motion of a free particle. Lagrangian coordinates. Dynamics: the fundamental principles of mechanics. Work and energy. Conervatives forces. The motion of a point under constraint. Discrete systems. Cardinal equations of dynamics. Non dissipative constraints. Lagrange's equations. Lagrange's equations for conservative systems. Conservations laws. One-dimensional motions. Qualitative analysis. Some classical problems: the problem of two bodies. Keplero's equations. Rigid body: Euler's angles. Angular velocity. Relative motions. Rigid body dynamics: inertia ellipsoid. Euler's equations. Lagrange's gyroscope. Equilibrium and stability: Lagrange-Dirichlet theorem. Instability criteria. Small oscillations. Variational principles of mechanics: Hamilton's principle. The Hamiltonian function (via Legendre transformation). Hamilton's equations. Canonical tranformations. Poisson brackets.
