Unit RATIONAL MECHANICS I

Course
Mathematics
Study-unit Code
55031206
Curriculum
In all curricula
Teacher
Francesca Di Patti
Teachers
  • Francesca Di Patti
Hours
  • 63 ore - Francesca Di Patti
CFU
9
Course Regulation
Coorte 2021
Offered
2023/24
Learning activities
Caratterizzante
Area
Formazione modellistico-applicativa
Academic discipline
MAT/07
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
Kinematics: kinematics of the point, kinematics of material systems and the rigid body, relative kinematics.
Dynamics: principles, statics and dynamics of the point, statics and dynamics of constrained systems.
Dynamics of material systems: geometry of the masses, Euler's equations, first integrals.
Analitica mechanics: D'Alembert's principle, Lagrange equations, Hamilton equations, Stability and small oscillations. Canonical transformations.
Reference texts
M. FABRIZIO, Elementi di Meccanica Classica, Zanicchelli, 2020.
H. GOLDSTEIN, C.P. POOLE, J.L. SAFKO, Classical Mechanics, III ed., Addison Wesley, 2001.
G. GRIOLI, Lezioni di Meccanica Razionale, Libreria Cortina, 2002.
V. I. ARNOLD, Mathematical Methods of Classical Mechanics, II ed., Springer-Verlag, 1989.
F. R. GANTMACHER, Lezioni di Meccanica Analitica, Editori Riuniti, 1980.
M. BRAUN, Differential Equations and their Applications, IV ed., Springer-Verlag, 1993.
Educational objectives
The course aims to give the students the fundamental mathematical instruments and methods useful for studying mechanical systems, in particular kinematics, statics and dynamics of a point particle and of the rigid body and Lagrangian and Hamiltonian mechanics. Students will become familiar with the mathematical modelling of mechanical systems and will learn the analytical methods useful to study the dynamics and statics of such systems.
Prerequisites
Basic knowledge of euclidean geometry, calculus, algebra and classical mechanics
Teaching methods
Lectures on all subjects of the course and respective exercises. Supporting material and a detailed program will be posted on unistudium.
Other information
It is recommended to attend the lectures.
Learning verification modality
Written and oral exam. The written test consists of 2 exercises of the same level of those solved during the classes. The pass mark for the written exam is 16/30. The oral exam can be postponed to any session within the end of the current academic year. The oral examination consists in an interview about 2/3 theoretical aspects. Operating time up to 30 minutes.

During the course, students can perform two optional intermediate tests. If in both tests they get a final mark greater or equal 16/30, they can skip the written exams and sit the oral exam.

For information about services for students with disabilities and/or DSA visit the page http://www.unipg.it/disabilita-e-dsa.
Extended program
KINEMATICS
Trajectory, velocity and acceleration. Relative kinematics and angular velocity vector. Addition of angular velocities. Velocity and acceleration fields of a rigid motion. Instantaneous axis of rotation. Planar rigid motions.

DYNAMICS OF THE SYSTEMS OF POINT MASSES AND RIGID BODIES
Kinetic energy, work, potential energy, kinetic energy Theorem and conservation of mechanical energy for systems of point masses. Cardinal equations of dynamics. Angular momentum and kinetic energy of a rigid body. Geometry of the masses: polar, axial and centrifugal moments of inertia. Matrix of inertia and ellipsoid of inertia. Principal axes of inertia. Rigid bodies rotating around a fixed axis.

DYNAMICS OF CONSTRAINED SYSTEMS
Ideal holonomic constraints for a system of point masses. Degrees of freedom, Lagrangian coordinates and tangent space. Principle of virtual works and its geometric interpretation. Symbolic equation of the dynamics and Lagrange equations of second kind. Lagrangian function. Quadratic structure of the kinetic energy. Hamiltonian formulation of the Lagrange equations. Generalized Dirichlet criterion. Quadratic approximation of the Lagrangian around a stable equilibrium and linearized equations of motion. Normal modes and frequency of the small oscillations around the stable equilibrium configuration.

LAGRANGIAN MECHANICS
Space of configurations and generalized coordinates. Holonomic and nonholonomic constraints. The principle of virtual work, and the principle of d'Alembert. The Lagrange equations, with or without non-conservative generalized forces.
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