00686 - Analytical Mechanics

Learning outcomes

After the course the student acquires the basic knowledge of the Lagrangian and Hamiltonian mechanics and on the
simple integrable models. In particular the student will be able to write the Lagrangian and the Hamiltonian
function for a mechanical systems, to study the phase space and the stability of the fixed points, to integrate
the equation of a central field and a rigid body with a rotational simmetry, to use variational principle
and canonical transformations.

Course contents

Dynamical systems: phase space, evolution equations, group properties, fisrt integrals, one dimensional problems, phase portraits, time law. Lagrangian mechanics: constraints and their realization, forces of constraint , geometry of curves and surfaces, D'Alembert's principle, generalized potentials, Lagrange and Hamilton's equations. Symmetries: Noether's theorem, scale and Galileo transformations. Mechanical models: harmonic and anharmonic oscillators, pendulum, central field with equation of orbits and Kepler laws, two body problem, Rotations and rigid body: Euler angles, velocity field, inertia matrix, principal axis, free motion and Poinsot cones, Lagrange top. Dynamics in a rotating frame. Equilibrium and its stability; small oscillations around a stable equilibrium. Hamiltonian mechanics: variational principles, canonical transformations, generating functions, Poisson brackets and Lie series, Maupertui's principle, Hamilton Jacobi equation and elements of canonical perturbation theory.

Readings/Bibliography

G. Turchetti Meccanica Classica dei Sistemi Fisici Ed. Zanichelli , Bologna

Landau L. D. Lifshitz Meccanica Ed. Boringhieri, Torino

Goldstein "Classical Mechanics" Addison- Wesley Pub. Co. A

Teaching methods

ex cathedra lessons

Assessment methods

Written and oral examination on the topics treated in the course

The texts of the written examination and the correspong solutions can be found in teh web site

http://www.physycom.unibo.it/MeccanicaAnalitica1.html

Office hours

See the website of Armando Bazzani