# 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 student is expected to acquire the skills to face and solve problems of dynamics of material systems. In particular, he is expected to become acquainted with the computation of centroids and inertia tensors of rigid bodies, and to be able to properly set the dynamical problem for a rigid body.

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

(1) Analysis and vector calculus: geometric vectors, vector product, orthogonal matrices, symmetric and antisymmetric operators, vector functions.

(2) 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.

(3) 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).

(4) 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

RANIERI ROLANDI

## 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 examination. The student can access the oral exam after he passes the written part.

Assessment methods

The written examination establishes that the student is able to set simple problems of rigid body dynamics. The oral assesses his understanding of the theory.

## 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.