Module page

Meccanica razionale

academic year:	2013/2014
instructor:	Camillo Cammarota
degree course:	Mathematics - DM 270/04 (triennale)
type of training activity:	caratterizzante
credits:	9 (72 class hours)
scientific sector:	MAT/07 Fisica matematica
teaching language:	italiano
program:	A-H
period:	II sem (03/03/2014 - 13/06/2014)

Lecture meeting time and location

Presence: highly recommended

Module subject: Mechanics of point-like particles. Unidimensional motions. Central motions, Kepler's laws. Cardinal equations of dynamics. Constrained systems. Velocity field for a rigid body. Dynamics of rigid bodies. Lagrange equations. Stability and small oscillations. Hamilton variational principle.

Detailed module subject: Kinematics of a point mass. Curves in R^3. Curvilinear abscissa. Unit vectors of the intrinsic triad. Speed and acceleration inherent to the intrinsic triad. Examples: cycloid, helical. One-dimensional equations of motions. First integral of energy. Qualitative analysis. The phase plane. Periodic motions. Estimate of the period. Small oscillations. Examples: harmonic oscillator; pendulum, isochronism of the cycloid motion. Mechanics of the point. Differential equation of motion. Example: motion of falling bodies. Equilibrium and stable equilibrium. Lyapunov theorem. Systems of points. Internal forces. Cardinal equations of dynamics. Motion of the center of gravity. Conservative forces. Work. Kinetic energy. First integral of energy. Example: two points interacting with the elastic force. Central forces. Polar coordinates and kinematic quantities. Equations of motion. First integrals of energy and angular momentum. Effective potential. Resolution by quadratures of motion and orbit. Case of gravitational interaction sun-earth. Binet formula. Deduction of Kepler's Laws: elliptical orbits, areolar velocity, third law. Earth-Moon system: the motion of the relative coordinate. Unidimensional constraints fixed and smooth. Projection according to the intrinsic triad . Equilibrium. Calculation of the reaction force. Examples: point mass on a plane curve subject to the weight and elastic force. Point mass on a surface. Example: spherical pendulum. Constrained mechanical systems. Example: double pendulum. Time-dependent and non-holonomic constraints. Lagrange equations. Configurations of equilibrium. Dirichlet's theorem. First integrals. Small oscillations around a stable equilibrium configuration. Normal modes. Kinematics of rigid bodies. Plane rigid motion. Instantaneous rotation center. Rotations and orthogonal matrices. Angular velocity. Fundamental formula of kinematics of rigid bodies. Mozzi' s axis. Rigid body mechanics. Angular momentum and kinetic energy. Theorem of Koenig. Inertia matrix. Ellipsoid of inertia and principal axes. Huygens' Theorem. Euler's angles. Expression of angular velocity in terms of Euler's angles. Euler's equations of a rigid body. First integrals. Special cases. Poinsot motion. Qualitative analysis. Lagrange equations of the spinning top: use of the first integrals; precession, nutation. Variational principle of Hamilton. Examples: bracristocrone, catenary. Relative dynamics: Coriolis acceleration.

Suggested reading:
E. Olivieri, Appunti di Meccanica Razionale, Aracne.
P. Buttà, P. Negrini, Note del corso di Meccanica Razionale, Ed. Nuova Cultura

Type of course: standard

Examination tests:

Knowledge and understanding: Successful students will be able to deal with topics concerning mathematical models of simple mechanical systems.

Skills and attributes: Successful students will be able to solve problems concerning mathematical models of simple mechanical systems.

Personal study: the percentage of personal study required by this course is the 65% of the total.

Statistical data on examinations