THREE-DIMENSIONAL MODELING
Course objectives
The course is devoted to introducing students to the methods of mathematical modeling, which is applied to a content pertaining to Classical Mechanics. This choice of subject is motivated by the interests of the Course of study. The program include a basic analysis of dynamical systems, of kinematics of a particle and of a single free rigid body, and then of the dynamics of systems composed of many rigid bodies, subjected to holonomic constraints. We specifically investigate equilibrium and its stability. Basic target of the course is the ability to analyse a simple problem of Rational Mechanics, in order to select the optimal strategy for its solution. Technically, we stress the Lagrangean formalism. Specific targets: A) Learning of basic knowledge of Rational Mechanics as a mathematical model of Mechanics. This target entails also basic notions of dynamical systems, solution of differential equations by matrix methods, notions of linear algebra and curves in space. B) Learning to set up and solve problems in Mechanics with a mathematical approach. Specifically, the student learns to translate principles of Physics into mathematical formalism, and viceversa to understand the applicative relevance of the predictions of the mathematical model. D), E) Development of the ability to understand qualitatively the solution and to exchange the results also answering simple questions, in order to seek help in textbooks or from experts.
Channel 1
ADRIANO BARRA
Lecturers' profile
Program - Frequency - Exams
Course program
System of ordinary differential equations, phase space, equilibrium and stability, application to mechanics.
The course treats the description and the prediction of the motion of rigid bodies, in the framework of Lagrangian mathematical formalism. We analyse in some detail the geometry of masses of rigid bodies and its relevance to motion, as well as the structure of the equations of motion. We also consider moving reference frames.
Motions as solutions to Ordinary Differential Equations (30 hours):
Examples of motions.
Ordinary Differential Equations and Systems.
Autonomous systems.
Equilibrium and stability.
The conservative case.
Representations in phase plane.
Kinematics (10 hours):
Change of reference frames.
Orthonormal triples.
Angular velocity.
Relative derivative in a moving triple.
Relative kinematics.
Coriolis theorem on accelerations.
Kinematics and dynamics of holonomic systems (25 hours):
Constraints.
Non-degeneracy conditions.
Independent coordinates and degrees of freedom.
Lagrangian coordinates.
Mechanical quantities in lagrangian coordinates.
Initial conditions.
Virtual displacements.
Heuristic introduction to constraints without friction.
Projection of the equation of motion on the tangent space.
The hypothesis of virtual work.
Lagrange equations.
Moving reference frames. Fictitious forces.
Lagrange equations in moving reference frames.
The conservative case. Lagrangian function.
Equilibrium and stability.
Rigid bodies (25 hours):
Local coordinates for rigid bodies.
The inertia tensor.
Principal axes.
Lagrange equations.
Lagrange equations in moving reference frames.
Motions of a rigid body with a fixed point.
Torque-free motion of a rigid body.
Rotations.
Cardinal equations.
Euler's equations.
Instantaneous axis of rotation.
Ruled surfaces of the motion.
Prerequisites
Important:
Elementary linear algebra.
Differential and integral calculus, in one and several real variables.
Curves and surfaces.
Ordinary differential equations, basic theory and techniques.
Indispensable:
exams of Analisi Matematica I and Geometria.
Books
E.M.N. Cirillo, Appunti delle Lezioni di Meccanica Razionale per l'Ingegneria, 2018, Compomat, Configni (RI).
D. Andreucci, E.N.M. Cirillo, Sistemi di equazioni differenziali ordinarie, 2025, CompoMat, Configni (Ri)
E.M.N. Cirillo, Appunti delle Lezioni di Meccanica Razionale per l'Ingegneria, 2018, Compomat, Configni (RI).
Frequency
Attending classes is voluntary and takes place in the classrooms and according to the schedule published by the Dean office. Attending classes, although warmly encouraged, does not contribute to the final mark.
Exam mode
The written exam consists in one test: the test is technical in nature and measures the student's capability of solving problems in Rational Mechanics, applying the methods and the techniques taught in the Course.
The oral exam is aimed at a more comprehensive assessment of the understanding of the fundamental structures of Rational Mechanics.
The exam is administered after the end of the course.
Evaluation criteria:
Minimal knowledge (grade 18-20);
average knowledge (grade 21-23);
sufficient capability of applying knowledge (grade 24-25);
good capability of applying knowledge (grade 27-28);
optimal capability of applying knowledge (grade 29-30 cum laude).
Bibliography
D. Andreucci, Meccanica Razionale. Modelli Matematici per l'Ingegneria, 2022.
H. Goldstein, C.P. Poole, J.L. Safko, Meccanica Classica, Zanichelli.
Lesson mode
Standard lectures devoted to teaching the theory.
Problem solving devoted to the development of the capability of applying the theory. Some problems are assigned as homework.
Learning is supported by the availability of the instructor in his office hours.
Channel 2
ADRIANO BARRA
Lecturers' profile
Program - Frequency - Exams
Course program
System of ordinary differential equations, phase space, equilibrium and stability, application to mechanics.
The course treats the description and the prediction of the motion of rigid bodies, in the framework of Lagrangian mathematical formalism. We analyse in some detail the geometry of masses of rigid bodies and its relevance to motion, as well as the structure of the equations of motion. We also consider moving reference frames.
Motions as solutions to Ordinary Differential Equations (30 hours):
Examples of motions.
Ordinary Differential Equations and Systems.
Autonomous systems.
Equilibrium and stability.
The conservative case.
Representations in phase plane.
Kinematics (10 hours):
Change of reference frames.
Orthonormal triples.
Angular velocity.
Relative derivative in a moving triple.
Relative kinematics.
Coriolis theorem on accelerations.
Kinematics and dynamics of holonomic systems (25 hours):
Constraints.
Non-degeneracy conditions.
Independent coordinates and degrees of freedom.
Lagrangian coordinates.
Mechanical quantities in lagrangian coordinates.
Initial conditions.
Virtual displacements.
Heuristic introduction to constraints without friction.
Projection of the equation of motion on the tangent space.
The hypothesis of virtual work.
Lagrange equations.
Moving reference frames. Fictitious forces.
Lagrange equations in moving reference frames.
The conservative case. Lagrangian function.
Equilibrium and stability.
Rigid bodies (25 hours):
Local coordinates for rigid bodies.
The inertia tensor.
Principal axes.
Lagrange equations.
Lagrange equations in moving reference frames.
Motions of a rigid body with a fixed point.
Torque-free motion of a rigid body.
Rotations.
Cardinal equations.
Euler's equations.
Instantaneous axis of rotation.
Ruled surfaces of the motion.
Prerequisites
Important:
Elementary linear algebra.
Differential and integral calculus, in one and several real variables.
Curves and surfaces.
Ordinary differential equations, basic theory and techniques.
Indispensable:
exams of Analisi Matematica I and Geometria.
Books
E.M.N. Cirillo, Appunti delle Lezioni di Meccanica Razionale per l'Ingegneria, 2018, Compomat, Configni (RI).
D. Andreucci, E.N.M. Cirillo, Sistemi di equazioni differenziali ordinarie, 2025, CompoMat, Configni (Ri)
E.M.N. Cirillo, Appunti delle Lezioni di Meccanica Razionale per l'Ingegneria, 2018, Compomat, Configni (RI).
Frequency
Attending classes is voluntary and takes place in the classrooms and according to the schedule published by the Dean office. Attending classes, although warmly encouraged, does not contribute to the final mark.
Exam mode
The written exam consists in one test: the test is technical in nature and measures the student's capability of solving problems in Rational Mechanics, applying the methods and the techniques taught in the Course.
The oral exam is aimed at a more comprehensive assessment of the understanding of the fundamental structures of Rational Mechanics.
The exam is administered after the end of the course.
Evaluation criteria:
Minimal knowledge (grade 18-20);
average knowledge (grade 21-23);
sufficient capability of applying knowledge (grade 24-25);
good capability of applying knowledge (grade 27-28);
optimal capability of applying knowledge (grade 29-30 cum laude).
Bibliography
D. Andreucci, Meccanica Razionale. Modelli Matematici per l'Ingegneria, 2022.
H. Goldstein, C.P. Poole, J.L. Safko, Meccanica Classica, Zanichelli.
Lesson mode
Standard lectures devoted to teaching the theory.
Problem solving devoted to the development of the capability of applying the theory. Some problems are assigned as homework.
Learning is supported by the availability of the instructor in his office hours.
- Academic year2025/2026
- CourseAerospace engineering
- CurriculumSingle curriculum
- Year2nd year
- Semester1st semester
- SSDMAT/07
- CFU3