Università di Pavia - Offerta formativa

MATHEMATICAL PHYSICS

Enrollment year

2020/2021

Academic year

2020/2021

Regulations

DM270

Academic discipline

MAT/07 (MATHEMATICAL PHYSICS)

Department

DEPARTMENT OF CIVIL ENGINEERING AND ARCHITECTURE

Course

CIVIL AND ENVIRONMENTAL ENGINEERING

Curriculum

PERCORSO COMUNE

Year of study

1°

Period

2nd semester (08/03/2021 - 14/06/2021)

ECTS

6

Lesson hours

60 lesson hours

Language

Italian

Activity type

WRITTEN AND ORAL TEST

Teacher

BISI FULVIO (titolare) - 6 ECTS

Prerequisites

Notions given in basic courses in Calculus Analisi Matematica), Linear Algebra, Geometry (Geometria e Algebra), and Physics (Fisica).

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. 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.

General kinematics

Center of mass of a system of material 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's equations of motion

Stability of motion

Stability of motion according to Ljapunov. Dirichlet-Lagrange theorem. First Ljapunov's instability criterion.

Normal modes

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

One-dimensional Continuum mechanics

Basic properties of curves. Unit tangent and unit normal vector to a plane curve; curvature of a curve. Intrinsic trihedron. Equilibrium equations for one-dimensional continuum bodies. Constitutive hypothesis: flexible and inextensible threads. Conservative active forces. Equilibriun profile of a homogeneous catenary. Suspended bridges.

Scalar and vector product; mixed product and repeated vector product; Diadics; symmetric tensors: spectral theorem. Skew-symmetric tensors: spin axis. Orthogonal tensors. 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.

General kinematics

Center of mass of a system of material 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's equations of motion

Stability of motion

Stability of motion according to Ljapunov. Dirichlet-Lagrange theorem. First Ljapunov's instability criterion.

Normal modes

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

One-dimensional Continuum mechanics

Basic properties of curves. Unit tangent and unit normal vector to a plane curve; curvature of a curve. Intrinsic trihedron. Equilibrium equations for one-dimensional continuum bodies. Constitutive hypothesis: flexible and inextensible threads. Conservative active forces. Equilibriun profile of a homogeneous catenary. Suspended bridges.

Teaching methods

Lectures (hours/year in lecture theatre): 22.5

Practical class (hours/year in lecture theatre): 37.5

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

Practical class (hours/year in lecture theatre): 37.5

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

Reccomended or required readings

F. Bisi, R. Rosso: Introduzione alla meccanica teorica (La Dotta, Bologna).

P. Biscari, C. Poggi, E.G. Virga, Mechanics Notebook (Liguori, Napoli).

P. Biscari, C. Poggi, E.G. Virga, Mechanics Notebook (Liguori, Napoli).

Assessment methods

Written test and oral exam (this latter is optional, and can be requested by the student or by the examiner). The student has to pass the test with 18/30 at least, and then, a few days later, he may take an oral test on theoretical topics. The final grade (including non-pass grade)

depends on the outcome of both tests.

depends on the outcome of both tests.

Further information

Written test and oral exam (this latter is optional, and can be requested by the student or by the examiner). The student has to pass the test with 18/30 at least, and then, a few days later, he may take an oral test on theoretical topics. The final grade (including non-pass grade)

depends on the outcome of both tests.

depends on the outcome of both tests.

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