After the course, students will be able to: (i) characterise the fundamental kinematic quantities which describe the motion of complex mechanical systems with a finite number of degrees of freedom and establish relationships between quantities evaluated in different frames of reference (both inertial and non-inertial); (ii) study the equilibrium configurations of such systems under the action of forces and analyse their stability; (iii) compute the reacting forces acting on these systems in equilibrium conditions; (iv) write the differential equations governing the motion of such systems; (v) study suitable approximations of the motion of these systems around stable equilibrium configurations. Students will also become familiar with the concepts of virtual work, power, energy and potential, which have broad applicability also beyond the realm of Rational mechanics.

Students are required to be familiar with concepts and methodologies from the courses of Calculus I, II and Geometry.

- Revision of vector calculus in R^3. Inner product and modulus, bases and components, change of basis, vector product, triple product, applied vectors, centre of parallel vectors. - Revision of differential geometry: curves and surfaces. Arc length, intrinsic reference system (tangent, normal and binormal unit vectors), parametrisation of a surface, tangent space. - Kinematics of a point particle. Equation of motion and trajectory, velocity and acceleration. - Kinematics of the rigid body. Rigidity constraint, rigid configurations, Poisson theorem and angular velocity, velocity and acceleration fields of a rigid motion, particular rigid motions (translational, roto-translational, rotational, planar), instant centre of rotation for planar motions, Chasles theorem. - Relative kinematics. Derivative of a vector with respect to two observers, Galileo theorem, Coriolis theorem, composition of angular velocities. - Constrained systems. Classification of the constraints (holonomic, nonholonomic, bilateral, unilateral, time-independent, time-dependent), Lagrangian coordinates and degrees of freedom, virtual velocities and displacements. - Geometry of areas and masses. Geometrical characteristics of planar sections, barycentre, centre of mass, statical moments of areas, moments of inertia, Huygens-Steiner theorem, matrix of inertia, principal axes of inertia. Examples of geometrical characteristics of special planar sections (rectangle, square, circle, sections composed by rectangles: T-section, double T-section, U-section). - Fundamental equations of Dynamics. Systems of forces, resultant and torque, work and virtual work of a force, work of forces on a holonomic system, generalised forces, work of forces on a rigid system, dynamics of a point particle, reactive forces, ideal constraints, linear momentum, angular momentum, first fundamental equation of Dynamics, second fundamental equation of Dynamics, angular momentum and its time derivative for rigid systems, Euler equations, first integrals. - Statics and equilibria. First fundamental equation of Statics, second fundamental equation of Statics, principle of virtual works, statics of rigid systems, statics of holonomic systems, equilibrium configurations, potential, principle of stationary potential, stability. - Kinetic energy. Koening theorem, kinetic energy for a rigid and a holonomic system, mass matrix, power of forces, theorem of the kinetic energy, principle of conservation of mechanical energy. - Lagrangian mechanics. Lagrange equations for holonomic systems, Lagrangian, first integral of canonical momenta. - Small oscillations. Linearisation of the Lagrange equations, characteristic frequencies of oscillation.

The course is organised in: - theoretical lectures (50 hours); - classroom exercises (30 hours). During classroom exercises students are exposed to the guided solution of problems focussed on the following topics: - kinematics of free and constrained point particles and rigid bodies; - relative kinematics; - calculation of the relevant geometrical characteristics of planar sections; - fundamental equations of dynamics; - fundamental equations of statics; - principle of virtual works, principle of stationary potential; - calculation of the equilibrium configurations of a mechanical system and discussion of their stability; - calculation of dynamical and static reactive forces; - Lagrange equations; - linearisation of the Lagrange equations about stable equilibrium configurations. During the last part of the course, classroom exercises will also focus on the solution of exercises from past exams.

Theory: - P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello, Meccanica Razionale, Springer, 2016. Exercises: - A. Muracchini, T. Ruggeri, L. Seccia. Esercizi e temi d'esame di Meccanica razionale, Esculapio, 2013. Further exercises will be provided, and partly solved, during practical classes. Moreover, a collection of solved exercises from past exams will be made available to the students through the course web site.

Lecture slides; Text book; Practice book; Exercises; Exercise with solutions ;

Exam: Written test; Compulsory oral exam;

The examination consists in a 120-minute written exam, which assesses the theoretical knowledge of the students and their ability to deal qualitatively with mechanical systems. The examination is composed of two parts: - a theoretical open-ended question (5 points); - an exercise on the topics taught during the lectures and the pratical classes (26 points). The maximum score obtainable from the written exam is 31, corresponding to the grade 30 with merit. During the written exam students may consult a form that they can fill in personally in advance not exceeding one side of an A4-size sheet. An oral exam, integrating the written one, is possible upon request of the students (if satisfactory in the written part) or of the teacher. The oral exam consists in theoretical questions, which may include both the proof of theorems and the application of solution techniques developed during the course. If requested, the oral exam contributes to the final grade of the examination together with the written exam. In particular, it can either increase or decrease the score obtained from the written exam.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.