MAT/07 (MATHEMATICAL PHYSICS)
DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING
1st semester (28/09/2020 - 22/01/2021)
60 lesson hours
WRITTEN AND ORAL TEST
Notions given in standard courses in Calculus and Geometry.
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.
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.
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.
Definition and main properties of the inertia tensor. Moments of inertia. Huygens-Steiner theorem. Theorem of perpendicular axes. Composition theorem. Material symmetry.
Balance equations. Kinetic energy theorem. Conservation laws. Power expanded in a rigid motion.
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.
Linearization of Lagrange's equations; normal co-ordinates. Oscillating, linear, and hyperbolic normal modes.
Lectures (hours/year in lecture theatre): 38
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.
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.
Sustainable development goals - Agenda 2030