# MATHEMATICAL PHYSICS

## AIMS AND CONTENT

AIMS AND LEARNING OUTCOMES

Course attendance and practice in class will allow the students to:

-understand the mathematical basis of the Newtonian kinematics and dynamics

- be able to solve problems involving the dynamics of point masses and rigid bodies.

Teaching methods

Frontal teaching, exercises. Attendance to the course is recommended.

SYLLABUS/CONTENT

**Elements of vector algebra and of the theory of geometric curves:**

Free and applied vectors. Vector quantities. Geometrical representation of free and applied vectors. Orthogonal projections. Scalar product. Orthonormal bases. Vector product. Mixed product, double vector product and their representation in components. Orthogonal matrices and change of orthonormal bases. Euler Angles. Linear operators and their representation in terms of matrices. Symmetric linear operators and their matrix representation. Symmetric and antisymmetric linear operators. Rectification formula. Arc length parameter. Frenet frame. Curvature and Torsion.

**Absolute kinematics:**

The concept of observer. Absolute space and absolute time axioms. Velocity, acceleration and their Cartesian representation.

**Relative Kinematics:**

Relative motion between frames. Angular velocity. Poisson formulas. Composition of angular velocities. Frame dragging motions. Theorem of composition of velocities and accelerations.

**Dynamics:**

First Principle of dynamics. Inertial mass. Momentum. Conservation of momentum for isolated systems. Second and third principle of dynamics. Work and Power of force. Conservative forces. Potential of conservative forces. Kinetic energy. Work-Energy theorem. Conservation of energy.

**Relative Dynamics:**

Fictious forces. Mechanics on the surface of Earth.

**Mechanics of point masses:**

Motion of a free point mass. Motion of a point mass along a smooth curve and a rough one.

**Mechanics of systems of particles:**

Systems of applied vectors. Resultant and total angular momentum of systems of vectors. Scalar invariant. Central axis. Reducible and irreducible systems of vectors. Center of a system of parallel applied vectors. Barycenter. Key mechanical quantities of systems of particles. Koenig theorem. Force-torque equations. Work-Energy theorem for systems of particles. Conservation laws of systems of particles.

**Mechanics of the rigid body:**

Reference frame comoving with a rigid body. Act of motion of a rigid body. Velocity and acceleration of the points of a rigid body. Examples of rigid motion. Composition of rigid motions. Key mechanical quantities of the rigid motion. Inertia tensor and its properties. Torque of a rigid body with respect to an axis. Inertia matrices. Huygens’ theorem and parallel-Axis theorem Force Torque equations for rigid bodies. Power of a system of forces acting on a rigid body. Work-Energy theorem for rigid bodies. Motion of a free rigid body. Ideal constraints applied to a rigid body. Pure rolling. Rigid body with a fixed axis. Poinsot motion and permanent rotations.

RECOMMENDED READING/BIBLIOGRAPHY

*The main topics of the course can be found in* Biscari P. et al. “Meccanica razionale”, Monduzzi editore (2007)--third edition. Lectures also contain elements of Massa E., “Appunti di meccanica razionale” (dipense); Grioli G. “Lezioni di meccanica razionale” Edizioni Libreria Cortina." Padua, Italy (1988); Demeio L. “Elementi di meccanica classica per l’ingegneria”, Città Studi edizioni (2016); Bampi F. , Zordan, C., “Lezioni di meccanica razionale” ECIG 1998; C. Cercignani, “Spazio, Tempo, Movimento”, Zanichelli; M.D. Vivarelli, “Appunti di Meccanica Razionale”, Zanichelli.

*Reference for exercises: *Muracchini A. et al. ”Esercizi e temi d’esame di Meccanica Razionale” (2013); Bampi F. et al “Problemi di meccanica razionale” ECIG, (1984).

## TEACHERS AND EXAM BOARD

**Ricevimento:** Monday 3pm - 6pm. It is strongly recommended to book in advance via email(sante.carloni@unige.it)/alualweb/teams.

**Ricevimento:** The teacher receives by appointment via email sent to cianci@dime.unige.it

Exam Board

SANTE CARLONI (President)

MARTA NAI RUSCONE

EVA RICCOMAGNO

ROBERTO CIANCI

## LESSONS

Teaching methods

Frontal teaching, exercises. Attendance to the course is recommended.

## EXAMS

Exam description

Written exam plus oral examination

Assessment methods

The oral examination will verify the acquisition of theoretical knowledge.

The written exam will assess the capability of the students to apply the knowledge acquired to sol the problem of the dynamics of point masses and rigid bodies.