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## MATHEMATICAL PHYSICS

CODE 60354 2023/2024 6 cfu during the 2nd year of 8715 INGEGNERIA CIVILE E AMBIENTALE (L-7) - GENOVA MAT/07 Italian GENOVA 1° Semester Prerequisites You can take the exam for this unit if you passed the following exam(s): Civil and Environmental Engineering 8715 (coorte 2022/2023) MATHEMATICAL ANALYSIS I 72290 GENERAL PHYSICS 56686 GEOMETRY 56716 Prerequisites (for future units) This unit is a prerequisite for: Civil and Environmental Engineering 8715 (coorte 2022/2023) HYDROLOGY & HYDRAULIC URBAN INFRASTRUCTURES 66097 STRUCTURAL ENGINEERING I 72543 STRUCTURAL MECHANICS II 66285 GEOTECHNICS 99062 URBAN PLANNING AND TRANSPORTATION ENGINEERING 84522 AULAWEB

## OVERVIEW

The course aims to provide the student with the basic elements of rational mechanics with particular reference to the mechanics of the rigid body and its applications to engineering.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The course provides the mathematical methods for describing mechanical systems. In particular the motion of systems with many degrees of freedom is studied the rigid body mechanics analyzed in details.

### AIMS AND LEARNING OUTCOMES

After the course completion the student is familiar with the statics and the dynamics of mechanical systems with finite degrees of freedom (particles systems and systems composed by rigid bodies).

### PREREQUISITES

mathematical calculus, basic linear algebra, mathematical analysis I, Physics I.

### 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

• Elements of Vector Algebra: Free and applied vectors. Vector quantities. Geometric representation of vector quantities. Vector structure of the space of free vectors. Scalar product of vectors. Orthonormal bases. Vector, triple scalar and triple vector product of vectors and their component representations. Orthogonal matrices. Change of orthonormal bases. Euler angles. Linear operators. Linear symmetric and skew-symmetric operators. Vector functions. Elements of geometric theory of a curve.
• Absolute Kinematics: Observer. Absolute Space and time. Frame of reference. Velocity, acceleration and their Cartesian and intrinsic representations. Rectilinear, uniform and uniformly accelerated motion. Circular motion. Harmonic motion. Ballistics problems. Central motions and Binet’s formula. Polar, cylindrical and spherical coordinates.
• Relative kinematics: Relative motion of frames of reference. Angular velocity. Poisson formulae. Theorem on composition of angular velocities. Transportation motion. Theorems on composition of velocities and accelerations.
• Dynamics: Newton’s first law. Inertial mass. Momentum of a particle. Momentum conservation for isolated systems. Newton’s second and third laws. Kinetic energy. Work and power of a force. Theorem of energy. Conservative forces. Potential of a conservative force. Theorem on conservation of energy.
• Relative Dynamics: Transportation inertial force. Coriolis inertial force. Earth Mechanics.
• Particle mechanics: Motion of a free particle. Friction laws. Motion of a  particle along a curve. Motion of a particle on a surface.
• Mechanics of systems: Systems of applied vectors. Resultant and resultant moment of a system of vectors. Scalar invariant. Central axis. Reducible and irreducible systems of vectors. Centre of parallel vectors and centre of gravity. Mechanical quantities of a system. Konig’s theorem. Momentum and angular momentum theorems. Theorem of energy for systems. Conservation laws for systems.
• Mechanics of a rigid body: The body-fixed reference frame of a rigid body. Rigid motion. Velocities and accelerations of the particles of a rigid body. Translational and rotational motions of a rigid body. Composition of rigid motions. Mechanical quantities of a rigid body. Inertia Tensor and its properties. Moment of a rigid body with respect to an Axis. Moments and products of Inertia. Inertia matrices. Huygens and parallel axes theorems. Momentum and angular momentum theorems for a rigid body. Power of a system of forces acting on a rigid body. Energy theorem for a rigid body. Motion of a free rigid body. Ideal constraints applied to a rigid body. Rotational motion of a rigid body about a fixed axis. Rotational motion of a rigid body about a fixed point. Poinsot motions. Elementary theory of a gyroscope and its application to the gyroscopic compass.

• Lecture notes provided on AulaWeb
• 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)

## LESSONS

### LESSONS START

https://corsi.unige.it/8715/p/studenti-orario

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## 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 oral examinations consists of three 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 on the website.

### 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.

### Exam schedule

Date Time Location Type Notes

### FURTHER INFORMATION

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.