00686 - Analytical Mechanics

Course Unit Page

Academic Year 2022/2023

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: definition of phase space, evolution equations, group properties of phase flows, fisrt integrals of motion, one dimensional problems, phase portraits, time law, linear dynamical systems. Definition of Equilibrium and study of its linear and non-linear stability. Lagrangian mechanics: covariant form of Newton equation, Minimum Action Principle, properties of LAgrangian equations, properties of Lagrangian equations, constraints and their realization, constraint forces and constraint dynamics , geometry of curves and surfaces, D'Alembert's principle, generalized potentials, Theory of Small Oscillations. Symmetries: Noether's theorem, search for first integrals of mortion and definition of integral system. Mechanical models: harmonic oscillators, spherical pendulum, central field with equation of orbits and Kepler laws, two body problem, double pendulum. Rotation group and rigid body: Euler angles, velocity field, inertia matrix, principal inertial axis, free motion and Poinsot cones, Lagrange top. Dynamics in a rotating frame. Hamiltonian mechanics: variational principles and phase space geometry, canonical transformations, generating functions, Poisson brackets and Lie series, Maupertuis' Principle, Hamilton Jacobi equation, elements of perturbation theory and Action-Angle variables.

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

Notes of the teacher

Teaching methods

ex cathedra lessons and exercises

Assessment methods

Written test composed by four exercices that allows to get a judgement of merit and to be admitted to the oral examination on the topics treated in the course. The judgement is considered a base for the final evaluation of the exam. In case of insufficient judgement it is not possible to take the oral examination.

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