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MATHEMATICAL PHYSICS

Enrollment year

2020/2021

Academic year

2020/2021

Regulations

DM270

Academic discipline

MAT/07 (MATHEMATICAL PHYSICS)

Department

DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING

Course

ELECTRICAL ENGINEERING

Curriculum

PERCORSO COMUNE

Year of study

1°

Period

1st semester (28/09/2020 - 22/01/2021)

ECTS

6

Lesson hours

60 lesson hours

Language

Italian

Activity type

WRITTEN AND ORAL TEST

Teacher

ROSSO RICCARDO (titolare) - 6 ECTS

Prerequisites

Notions given in standard courses in Calculus and Geometry.

Learning outcomes

The course aims at giving an overwiev of classical mechanics to show that an adequate mathematical formulation can give a deep insight into the problems of this discipline.

Course contents

Vector and tensor algebra

Scalar and vector product; mixed product and repeated vector product; Diadics; symmetric tensors: spectral theorem. Skew-symmetric tensors: spin axis. Orthogonal tensors: Euler's angles. Systems of vectors

Relative and rigid-body kinematics

Poisson formulae; Time derivatives of vectors in different frames. Basic formulae in relative kinematics. Fundamental formula in rigid kinematics; Planar rigid motion: Chasles theorem.

General kinematics

Center of mass of a system of mateiral points; Momentum, moment of momentum, and kinetic energy. Transport theorem for moment of momentum. König's theorem.

Inertia tensor

Definition and main properties of the inertia tensor. Moments of inertia. Huygens-Steiner theorem. Theorem of perpendicular axes. Composition theorem. Material symmetry.

General dynamics

Balance equations. Kinetic energy theorem. Conservation laws. Power expanded in a rigid motion.

Lagrangian dynamics

Lagrange equations. Cyclic coordinates and conservation laws.

Rigid body dynamics

Euler's equations. Poinsot case. Lagrange's top.

Stability of motion

Stability of motion according to Ljapunov. Dirichlet-Lagrange theorem. First Ljapunov's instability criterion. Stability of steady rotations in Poinsot motions.

Normal modes

Linearization of Lagrange's equations; normal co-ordinates. Oscillating, linear, and hyperbolic normal modes.

Scalar and vector product; mixed product and repeated vector product; Diadics; symmetric tensors: spectral theorem. Skew-symmetric tensors: spin axis. Orthogonal tensors: Euler's angles. Systems of vectors

Relative and rigid-body kinematics

Poisson formulae; Time derivatives of vectors in different frames. Basic formulae in relative kinematics. Fundamental formula in rigid kinematics; Planar rigid motion: Chasles theorem.

General kinematics

Center of mass of a system of mateiral points; Momentum, moment of momentum, and kinetic energy. Transport theorem for moment of momentum. König's theorem.

Inertia tensor

Definition and main properties of the inertia tensor. Moments of inertia. Huygens-Steiner theorem. Theorem of perpendicular axes. Composition theorem. Material symmetry.

General dynamics

Balance equations. Kinetic energy theorem. Conservation laws. Power expanded in a rigid motion.

Lagrangian dynamics

Lagrange equations. Cyclic coordinates and conservation laws.

Rigid body dynamics

Euler's equations. Poinsot case. Lagrange's top.

Stability of motion

Stability of motion according to Ljapunov. Dirichlet-Lagrange theorem. First Ljapunov's instability criterion. Stability of steady rotations in Poinsot motions.

Normal modes

Linearization of Lagrange's equations; normal co-ordinates. Oscillating, linear, and hyperbolic normal modes.

Teaching methods

Lectures (hours/year in lecture theatre): 38

Practical class (hours/year in lecture theatre): 22

Practicals / Workshops (hours/year in lecture theatre): 0

Practical class (hours/year in lecture theatre): 22

Practicals / Workshops (hours/year in lecture theatre): 0

Reccomended or required readings

F. Bisi, R. Rosso: Introduzione alla meccanica teorica.

Assessment methods

Written test and oral exam. The student has to pass the test with 18/30 at least, and then, a few days later, he will take an oral exam on theoretical topics. If a student passes the written test, he can decide to avoid the subsequent oral test. In that case, however, he will obtain his mark, whenever it does not exceed 21/30 while, if the written exam exceeds this threshold and the student does not take the oral exam, his mark will be 21/30.

Further information

Sustainable development goals - Agenda 2030