Teacher

prof. aggr. Vittorino TALAMINI

Credits

6 CFU

Language

Italian

Objectives

The course deals with the classical mechanics of systems of mass points and introduces the Lagrange formalism.

Acquired skills

- Kinematics of rigid bodies.
- Dynamics of systems with a finite number of degrees of freedom.
- Techniques for the qualitative study of ordinary differential equations.

Lectures and exercises (topics and specific content)

Basic vector algebra: scalar and vector product; vector identities; gradient (2 hours).
Particle kinematics: reference frames; law of motion; velocity and acceleration; trajectory; intrinsic basis and Frenet's formulas (5 hours).
Kinematics of rigid bodies: angular velocity and Poisson's formulas; velocity field in a rigid motion; Euler angles (4 hours).
Relative kinematics: composition of velocities, accelerations, and angular velocities (3 hours).
Foundations of particle dynamics: Newton's first law; inertial frames; Newton's second law; phase space (4 hours).
Conservative systems with one degree of freedom: integral curves; phase curves; potential and mechanical energy; qualitative study of motion; reduction to a first-order problem (4 hours).
Oscillating systems with one degree of freedom: harmonic oscillator; period of oscillations around a configuration of stable equilibrium; simple pendulum; dissipative systems and damped harmonic oscillator; forced oscillations and resonance (5 hours).
Motion in three dimensions: conservative fields of forces; constraints; motion of a particle along a smooth curve; inertial forces (5 hours).
Foundations of dynamics for systems: equations of motion for a system; external and internal forces; work and conservation of energy; Newton's third law; linearity and superposition (4 hours).
Dynamics of systems: centre of mass; global equations and conservation laws; kinetic energy and angular momentum; Koenig's theorems (5 hours).
Rigid bodies: kinetic energy and angular momentum for a rigid body; inertia operator, inertial matrix, and inertia moments; principal directions; power for a rigid body; Euler's equations; force-free motions (9 hours).
Foundations of Lagrangian dynamics: holonomic systems with perfect constraints; D'Alembert's principle; Lagrange equations; kinetic energy for a holonomic system; kinetic moments; cyclic coordinates and constants of motion (10 hours).
Exercises (20 hours).

References

- C. Cercignani, Spazio tempo movimento (Zanichelli, Bologna, 1976)
- M. Fabrizio, Elementi di meccanica classica (Zanichelli, Bologna, 2002)
- A. Fasano e S. Marmi, Meccanica analitica (Bollati Boringhieri, Torino, 1993)
- L. D. Landau e E. M. Lifshits, Meccanica (Editori Riuniti, Roma, 1976)

Type of exam

Written and oral

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