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Second Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2017]  PERCORSO COMUNE 8 64


1) Knowledge and understanding skills:
Teaching for students of the II year of the Bachelor's Degree in Mathematics is ideally divided into three parts: The first part aims to provide the student with theoretical and operational knowledge of rigid kinematics concepts, including changes necessary when talking about relative kinematics; in the second part it deals with the dynamics of the material point with some practical applications and begins to describe the concept of rigid bodies subject to constraints; In the third part we study the dynamics of rigid bodies, the concept of equilibrium configuration, and the analytic criteria needed to determine the equations of motion and equilibrium positions. The various topics are described in detail by providing a rigorous theoretical justification.
2) Knowledge and understanding skills applied:
During the course, the possible applications of the methods studied will be discussed, both with regard to the resolution of particular mathematical problems and their application in related disciplines (physics and engineering).
3) Autonomy of judgment:
The course provides the diligent students with the skills necessary to apply the mathematical methodologies acquired to solve the application problems that typically occur in the mechanics.
4) Communicative Skills:
The oral examination takes into account the student's ability to expose the topics studied neatly and consistently
5) Learning ability:
The course provides diligent students with enough preparation to model any other problem in applied mathematics.


It is required the knowledge of the arguments of Mathematical Analisys , that is the methodology of studying real. Moreover, are requested elements of analytic geometry and methods to solve linear algebraic systems (Cramer and Rouchè-Capelli ‘ s Theorems) with all the notions lying behind these (Matrices, determinants, etc.).


Vectorial spaces: Elements of vectorial calculus. Vectorial equations. Scalar Invariant, vectorial invariant and central axis. Definition and properties of the centre of a system of applied and parallel vectors with a non zero sum. Elements of the properties of regular curves and surfaces, for their applications to Mechanics.
Kinematics: Axiom of absolute time. Velocity and acceleration. Circular motion, armonic motion, composed motions, helicoidal motion. Rigid motions, fundamental formula of rigid motions, angular velocity, Poissons ‘ s formulae. Eulero ‘ s angles. Axis of istantly rotation. Rigid motions with zero kinematic invariant. Planar motions, centre of istantly rotation, areal velocity. Instantly motion, Mozzi ‘ s Theorem. Relative motions, principle of relative motion. Coriolis ‘ s Theorem, relation between absolute and relative time derivative, composition of angular velocities. Regular precessions. Pure rolling motion.
Dynamics: Principles of Mechanics. Determinism of Mechanics for a free point. Equilibrium. Relative Dynamics, dragging forces and Coriolis’ s force, relative equilibrium condition, weight, terrestrial dynamics. Heavy point’ s motion, securety parable, deviation of heavy points towards est, heavy points motion in a fluid, critical velocity. Generality on constraints. Finite, possible and virtual shifting. Lagrangian coordinates. Elementary work . Principle of constraints reactions. Equations of the motion: Cardinal equation and Lagrange ‘ s Equations. Theorem of momentum, theorem of angular momentum, second form of Lagrange ‘ s Equations. Centere of Mass, Theorem for the motion of the Center of Mass. Lagrangian expression of the kinetic energy and determinism of the Lagrange ‘ s equations. Inertial Matrix. Generalized Hyghens ‘ s Theorem. Centers of Mass and inertial tensors for some homogeneous figures.. Angular momentum for a rigid body with a fixed point O, for rigid bodies without fixed points, König Theorem. Equlibrium conditions, both lagrangian or through the balance equations.
Study of the motion: Systems of linear differenzial equations with constant coefficients, motion in a first approximation. First Integrals. Weierstrass‘ s discussion.
Conservative forces and conservative strains , Potentials. Some useful potentials. Balance integral of total energy.

Teaching Methods

Slides are used to espose the principal arguments, while the teacher furnishes more particulars with the classical method. It is possible, for a part of the time and for well disposed students, to excercise themselves in front of the other students.

Verification of learning

The examination consists in an oral text. Each student may choose to subdivide it in three distinct sections. An argument is extracted for both of them from a list previously furnished to students. Votes are assigned using the following method:

Votes 1-10: Evaluate the student's (both theoretical and operational) knowledge of the concepts of rigid kinematics, both in an absolute reference and in a relative one.

Votes 1-10: We evaluate the student's ability to know and apply the methods studied, both with regard to the resolution of particular problems of material dynamics and rigid rigid systems, as well as their application in related disciplines (physics and engineering).

Votes 1-10: We evaluate the student's ability to acquire and apply the methods to find equations describing motion and equilibrium. It is also taken into account the student's ability to expose these topics in an orderly and coherent manner and to have acquired the ability to model any other problem in applied mathematics.

The final vote will be the sum of these 3 partial votes with the addition of 1; if the total exceeds 30, the praise is also given.

During the period of lessons, two occasions are scheduled to do the formative evaluation, as suggested in the course Discentia.


Lectures furnished by prof. Sebastiano Pennisi

Lezioni di Meccanica Razionale, Prof. Salvatore Rionero, Liguore Editore.

Meccanica Razionale, Biscari P., Ruggeri T.,Saccomandi G. Vianello M.,
Unitext, volume 69, Springer

Esercizi e temi di Meccanica Razionale, Muracchini A., Ruggeri T., Seccia L.
Società Editrice Esculapio

• T. Ruggeri, Appunti di Meccanica Razionale: Richiami di Calcolo Vettoriale e Matriciale, Ed. Pitagora, Bologna.

Complementi ed esercizi di Meccanica Razionale, Mauro Lo schiavo, Società Editrice Esculapio

More Information

Lesson notes and other tools are shared with students via an updated Dropbox folder year after year.

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