Learning outcomes of the course unit
The goal is to give the foundations and the principal results of a rigorous mathematical approach to the Mechanics of the mass point and of the rigid body.
A good knowledge of the contents of the courses of Analisi AB, Geometria and Fisica AB is almost necessary.
Course contents summary
Geometry. Vector spaces: linear combination, generators, bases, dimension, coordinates.
Linear applications: definition, eigenvalues, eigenvectors.
Analysis. Functions on Rn: limits, differentiation, integration, differential equations.
Mathematical aspects of Classical Mechanics.
The euclidean space E3 of dimension 3 and its modelling vector space V3 of free
vectors. Isometries. Applied vectors and systems of applied vectors. Regular curves
in E3, tangent and normal vectors, curvature. Systems of coordinates in E2 and E3.
Space, time and observers: the absolute time function. Frames of reference. Absolute
Kinematics: motions, equations of motions. Direct and inverse problems in Kinematics,
central motions. Relative Kinematics: angular velocity, rigid Kinematics,
Galilei’s and Corioli’s theorems. Contact motion and pure rolling.
Statics and Dynamics of a point particle.
Mass, Inertia Principle, inertial observer. The three laws of Dynamics in an inertial
frame and in a generic frame. Conservative forces and their potential.
Statics and Dynamics of the free particle. Energy balance. Statics and Dynamics
of constrained particle. Constitutive characterization of a constraint. Friction laws.
Stability of the equilibrium.
Mass and mass density, center of mass, total momentum, angular momentum, kinetic
energy. Koenig theorem. Concentrated and distributed forces. Conservative forces.
General formulation of the Equations of motion of particle systems.
Statics and Dynamics of the rigid body.
Rigid body: total momentum, angular momentum and kinetic energy. Inertia tensor
and inertia momentum with respect to an axis, principal moments of inertia. The
equations of motion for the rigid body. Free rigid body. Constitutive characterization
of constraints acting on a rigid body. Rigid body with fixed point and with fixed
axis. Examples of motions.
Additional arguments and deeper investigations.
Statics and Dynamics of systems of linked rigid bodies.
Stability of the equilibrium for rigid bodies and for systems of linked rigid bodies.
Arguments of Lagrangian Mechanics.
Material prepared by the teacher.