LEARNING OUTCOMES

Formative aims

A rigorous approach to Newtonian mechanics with applications to the motion of rigid systems. The basic methods of analytical mechanics to solve equilibrium and dynamic problems.

Technical skills

Ability to write potential and kinetic energies of mechanical systems with few degrees of freedom, to  derive the differential equations of motion and the equilibrium conditions.  

AIMS AND LEARNING OUTCOMES

The main objective of this module is a rational approach to the following issues:

1) Kinematics of matererial point by a geometrical description of spatial curves.

2) Equilibrium and dynamics of discrete or continuum material systems using cardinal equations of mechanics

3) Inertial èproperties of material systems

4) A Lagrangian description by the introduction of free coordinates for system subject to constraints and the role of first integrals.

5) Equilibrium and stability by analytical approaches.

The module aims to give some technical skills on the following problems:

1) KInematical and dynamical description of a system subject to constraints

2) Computation of kynetic anf potential enrgies by the Lagrangian formalism and the derivation of differential equations of motion

3) Computation of equilibrium configurations of a mechanical system and a discussion on their stability.

At the end of the course the student can arrive at the following results:

1) The knowledge of the algebraic and analytical tools necessary to the description of motion.

2) Understanding the main mathematical techniques relating linear momentum, angular momentum and energy to the inertial and dynamical properties of a mechnical system.

3) The ability to analize a mechanical systems subject to given loads and constraints, achieving results on the equilibrium conditions and obtaining the differential equations of motion, also recognizing possible first integrals.

TEACHING METHODS

Lectures on the theoretical contents with applications and exercises.

SYLLABUS/CONTENT

INTRODUCTION

MASSIVE POINT

• Kinematics of the massive point
• Mechanics of the free and constrained point

RELATIVE MECHANICS 

•     Derivation and observer, Poisson formula
•     Relative kinematics
•     Relative mechanics

DISCRETE SYSTEMS

•   Newton third principle and internal forces
•   Equation for the cinetic and angolar momenta
•   Center of mass

RIGID BODY

•     RIgidity constraints and the law of distribution of velocities
•     Kinematics
•     Operator of inertia
•     Mechanics of the rigid body
•     Constrained rigid body   

ANALITICAL MECHANICS 

•     Olonomous systems
•     D'Alembert principle
•     Euler-Lagrange equation
•     Eulero-Lagrange equation and cardinal equations.

    
INTRODUCTION TO STABILITY THEORY

•    Equilibrium and stability for mechanical systems
•    Small oscillations

RECOMMENDED READING/BIBLIOGRAPHY

• Lecture notes by the teacher
• Bampi F., Benati M., Morro A., Problemi di Meccanica Razionale, Ecig (Genova)
• Bampi F. e Zordan C., Lezioni di Meccanica Razionale, Ecig (Genova)
• Goldstein H., Meccanica Classica, Zanichelli (Bologna, 1971)
• Levi M., Classical mechanics with calculus of variations and optimal control - An intuitive introduction. AMS (USA, 2014)