Lecturer: Riccardo Rosso  

Course name: Mathematical Physics
Course code: 500474
Degree course: Ingegneria Civile e Ambientale
Disciplinary field of science: MAT/07
L'insegnamento costituisce attività di base per: Ingegneria Civile e Ambientale
University credits: ECTS 6
Course website: n.d.

Specific course objectives

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 programme

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.

Course entry requirements

Notions given in standard courses in Calculus (Analisi A and B) and Geometry.

Course structure and teaching

Lectures (hours/year in lecture theatre): 60
Practical class (hours/year in lecture theatre): 0
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

Lecture notes available at the course website.

R. Rosso. Esercizi e Complementi di Meccanica Razionale. CUSL.

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

Testing and exams

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.