
Course unit
ANALYTICAL MECHANICS
IN02105695, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2016/17
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Basic courses 
MAT/07 
Mathematical Physics 
9.0 
Mode of delivery (when and how)
Period 
First semester 
Year 
2nd Year 
Teaching method 
frontal 
Organisation of didactics
Type of hours 
Credits 
Hours of teaching 
Hours of Individual study 
Shifts 
Lecture 
9.0 
72 
153.0 
No turn 
Start of activities 
25/09/2017 
End of activities 
19/01/2018 
Prerequisites:

The course contents (1) MATHEMATICAL ANALYSIS 1
and (2) INTRODUCTION TO LINEAR ALGEBRA AND GEOMETRY 
Target skills and knowledge:

Basic training in classical Rational Mechanics and in Mechanics of Lagrangian systems.
Knowledege of some basic facts in the dynamics of rigid bodies with applications of interest in mechanical engineering. 
Examination methods:

Traditional 
Assessment criteria:

Written examination and oral examination 
Course unit contents:

Classical Rational Mechanics, Lagrangian Mechanics, dynamics of rigid bodies. In details:
Vectors recalls. Cartesian components; scalar, vectorial and mixed product; double vectorial product.
Vector fields. Torsors; central axis; sum of torsors.
Applied vector systems. Polar moment, axial moment; equivalence of two systems of applied vectors; elementary operations and reducibility; reduction of an applied vector systems; plane vector systems; parallel vector systems.
Point kinematics. Vector velocity and acceleration in FrenetSerret (intrinsic) frame; elementary displacement.
Kinematics of rigid systems. Rigid displacements and rigid motions, cartesian equations of a rigid motion, expression of angular velocity; Poisson's formulas; velocity, acceleration and elementary displacement fields; elementary rigid motions; plane and spherical rigid motions; motion acts and their compositions.
Relative kinematics and applications to rigid motions. Velocity composition theorem; Coriolis theorem, geometric description of a rigid motion as mutual rolling of two surfaces; polar trajectories in plane rigid motions; precessions.
Mass geometry. Mass; mass center of a discrete or continuous system; inertial momentum; HuygensSteiner theorem; inertial tensor and ellipsoid of inertia; gyroscope.
Mass kinematics. Linear momentum; angular momentum; kinetic energy; motion relative to the center of mass; Koenig theorem; expressions of kinetic energy and angular momentum for a rigid body.
Kinematics of constrained systems. Geometric, kinematic, bilateral, unilateral constraints; holonomic systems; possible and virtual displacements.
Forces and Work. Positional and conservative force fields; elementary and effective work; work along a finite path for positional and conservative forces; work of a force system; expression of work of a force system acting on a rigid body; possible and virtual works in holonomic systems.
Principles of mechanics. Inertial frames and Newton laws for discrete systems; postulate of reaction forces; cardinal equations of dynamics for physical systems formed by concentrated masses and rigid bodies with internal and external constraints; kinetic energy theorem and integral of energy; noninertial frames and apparent forces.
Statics of a system of material points. Equilibrium positions; principle of virtual works for a system of material points; equilibrium in a noninertial frame.
Statics of the rigid body. Equilibrium configurations, cardinal equations of statics and principle of virtual works for a rigid body.
Lagrangian Mechanics for holonomic systems. General equation of dynamics; free coordinates; Lagrange equations in the first and second form; equilibrium stability and small oscillations in the neighbourhood of a stable equilibrium configuration; normal modes; stability criterion of LagrangeDirichlet and instability criterion of Lyapounov.
Dynamics of spherical rigid motions. Eulero's equations; gyroscopic phenomena, study of the Poinsot motions and their geometric description.
Applications of the theory to discrete systems and rigid bodies. Comparison of simple pendulum and compound pendulum; staticdynamic comparison of reaction forces on the rotation axis of a rigid body. 
Planned learning activities and teaching methods:

Learning activities: regular exercises assigned for home, quiz/test online on the page Moodle of the course, classroom exercises to check on the degree of understanding of the arguments.
Teaching methods: lectures; analysis of the solutions to the exercises in the classroom proposals. 
Additional notes about suggested reading:

Moodle page of the course where resources are accessible as the daily diary of the lessons, exercises, the examination carried out with solutions, slides of individual lessons. 
Textbooks (and optional supplementary readings) 

A. MONTANARO, Meccanica Razionale. Padova: Librerie Progetto, 2017. Nuova edizione

A. MONTANARO, Problemi di Meccanica Razionale. Padova: Edizioni La Dotta  Bologna, 2017. Nuova edizione


