# MATHEMATICAL PHYSICS 1

## AIMS AND CONTENT

LEARNING OUTCOMES

The module aims to provide knowledge of mechanics of multi -degree of freedom systems . The case of the rigid body e'trattato in detail .

AIMS AND LEARNING OUTCOMES

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

Teaching methods

60 standard teaching hours

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.

**Mechanics of a particle:**

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.

**Outlines of Lagrangian Mechanics:**

Principle of the stationary potential for the equilibrium of a conservative holonomic system (without proof). Lagrange equations for a conservative holonomic system (without proof) .

RECOMMENDED READING/BIBLIOGRAPHY

Enrico Massa, Elementi di Meccanica Razionale, dispense Università di Genova.

T. Levi-Civita and U. Amaldi, Lezioni di Meccanica Razionale, Zanichelli, Bologna (1984).

B. Finzi, Meccanica Razionale, Vol. II, Zanichelli, Bologna, (1965).

G. Grioli, Lezioni di Meccanica Razionale, Edizioni Libreria Cortina, Padova, (1985).

P. Biscari, T. Ruggeri. G. Saccomandi and M. Vianello, Meccanica Razionale per l'Ingegneria, Monduzzi Editore S.p.A., Bologna, (2008)..

## TEACHERS AND EXAM BOARD

Exam Board

DANILO PERCIVALE (President)

STEFANO VIGNOLO (President Substitute)

## LESSONS

Teaching methods

60 standard teaching hours

## EXAMS

Exam description

A written and an oral test, after passing the written test with a mark greater then or egual to 16.

Assessment methods

A written and an oral test.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

14/06/2021 | 14:30 | LA SPEZIA | Scritto | |

18/06/2021 | 10:00 | LA SPEZIA | Orale | |

05/07/2021 | 14:30 | LA SPEZIA | Scritto | |

09/07/2021 | 10:00 | LA SPEZIA | Orale | |

13/09/2021 | 14:30 | LA SPEZIA | Scritto | |

17/09/2021 | 10:00 | LA SPEZIA | Orale |