# MATHEMATICAL PHYSICS

OVERVIEW

The mathematical tools acquired in the courses of Analysis and Geometry are used for a precise and systematic formulation of the mechanics of material systems with particular emphasis for rigid body mechanics.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course provides the mathematical skills to face and solve the dynamical problem for material systems, and, in particular, to properly set the dynamical equations for a constrained rigid body

AIMS AND LEARNING OUTCOMES

The kinematics and dynamics of material systems are presented in a systematic and precise way, using the mathematical methods acquired in the courses of analysis and geometry.

Particular attention is paid to the formulation of rigid body mechanics through the use of cardinal equations.

Teaching methods

The course includes lectures at the blackboard in which the topics of the program are presented. Examples and exercises, designed to clarify and illustrate the concepts of the theory, are also carried out.

SYLLABUS/CONTENT

- Analysis and vector calculus: geometric vectors, vector product, orthogonal matrices, symmetric and antisymmetric operators, vector functions.
- Complements of kinematics and dynamics of a particle: reference systems, Euler angles, Poisson's formulas, basic kinematic concepts and their representation in Cartesian, polar, spherical and cylindrical coordinate systems.
- Kinematics and dynamics of material systems: systems of applied vectors, center of mass, kinetic energy and angular momentum for a system of particles, Koenig theorem, internal and external forces, work, power, energy, conservative systems, cardinal equations, constrained systems (rudiments).
- Rigid body: definition and degrees of freedom, fixed reference frame, rigid motions, angular momentum and kinetic energy, inertia tensor, transposition theorem, material symmetries and principal axes of inertia, equations, permanent rotations, Poinsot motions, constrained systems (outline)

RECOMMENDED READING/BIBLIOGRAPHY

- Lecture notes by the teacher
- Bampi Zordan “Meccanica Razionale. Con elementi di probabilità e variabili aleatorie” ECIG (2003)
- Goldstein “Classical Mechanics”, Addsion-Wesley; 3 edition (2001)
- Fasano, Marmi, Pelloni “Analytical Mechanics” Oxford Uiversity Press (2006)

## TEACHERS AND EXAM BOARD

**Ricevimento:** Appointment on student's request (send an email to carm@sv.inge.unige.it).

Exam Board

OTTAVIO CALIGARIS (President)

CLAUDIO CARMELI (President)

MAURIZIO SCHENONE

## LESSONS

Teaching methods

The course includes lectures at the blackboard in which the topics of the program are presented. Examples and exercises, designed to clarify and illustrate the concepts of the theory, are also carried out.

## EXAMS

Exam description

The exam consists of a written and an oral exam. The student must pass the written test and, after, he can access to the oral exam.

The written exam lasts three hours and consists of two problems on the subjects covered in class. The first problem is about the determination of barycenters and inertia tensors. The secondo problem is about the determination of the equations of motion of a constrained rigid body. The marks of the written tests are A(=excellent),B(=good),C(=sufficient),D=(not sufficient). The test is passed if the mark is A,B,C.

The oral examinations consists of four questions on the topics of the course.

Both during the written exam and during the oral exam, it is not possible to consult anything or use the calculator (except for the exceptions foreseen for students with DSA).

To participate in any type of test, students must register at least two days before the date of the exam on the website

https://servizionline.unige.it/studenti/esami/prenotazione

Assessment methods

The written test verifies that the student knows how to set and solve problems of kinematics and dynamics of material systems and in particular of mechanics of the constrained rigid body.

The oral test verifies that the student has overcome any gaps that emerged in the written test and has acquired the theoretical technical skills.The quality of the presentation, the correct use of the specialized lexicon and the critical reasoning skills contribute to the final evaluation.

## FURTHER INFORMATION

Pre-requisites :

Although the course provides an introductory part, it is appropriate that the student is familiar with: linear algebra (vectors and linear transformations), derivation and integration, kinematics and dynamics of a material particle.