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