MATHEMATICAL PHYSICS
Stampa
Enrollment year
2020/2021
2020/2021
Regulations
DM270
MAT/07 (MATHEMATICAL PHYSICS)
Department
DEPARTMENT OF CIVIL ENGINEERING AND ARCHITECTURE
Course
CIVIL AND ENVIRONMENTAL ENGINEERING
Curriculum
PERCORSO COMUNE
Year of study
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.
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