00686 - Analytical Mechanics (M-Z)

Learning outcomes

Upon completion of the course, the student possesses basic knowledge of Lagrangian and Hamiltonian mechanics and the main integrable models. In particular, the student is able to write the Lagrangian and Hamiltonian functions of a mechanical system; analyze the phase space for one-dimensional systems; recognize the existence of prime integrals of motion related to symmetries of the system; study the stability of equilibria and solve the equations of motion in the approximation of small oscillations; discuss the solutions of the equations of motion for the central field and the spinning top; use variational principles of least action to write the equations of motion; and use perturbative methods to study mechanical systems.

Course contents

- Principles of Relativity and Determinism: Galileo's Group and Newton's Equations

- Equations of Motion: systems with one and two degrees of freedom

- Conservative Field of Forces and the Momentum of the Quantity of Motion

- Motion in a central field: Kepler's laws

- Variational Principles: Lagrange's and Hamilton's equations

- Legendre's Transformation and Liouville's Theorem

- Lagrange mechanics on varieties: holonomic constraints and Lagrangian dynamical systems

- Theorem ofE.Noether

- D'Alembert's Principle

- (small) Oscillations:eigenfrequencies and parametric resonance

- the Rigid Body: composition of motions, force of inertia, Coriolis force, Euler equations

- Canonical Formalism

- Integrable systems

Readings/Bibliography

• VI Arnold "Mathematical Methods of Classical Mechanics"

This is a rather advanced textbook on the subject, but it will be followed at a pace suitable for the students of the course, simplifying and integrating the various topics. Other reference texts (in order of importance for this course) are:

• Instructor's Notes "Arnold for dummies" (Degli Esposti, Graffi, Isola)

and also the open resource for students:

• "Introduction to Dynamical Systems Vol.2"

Teaching methods

Frontal lectures: blackboard + slides

Assessment methods

Written and Oral Evidence

Teaching tools

Students with Specific Learning Disabilities (SLD) or temporary or permanent disabilities are advised to contact the University office in charge in advance (https://site.unibo.it/studenti-con-disabilita-e-dsa/en ). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, who will assess their appropriateness in relation to the learning objectives of the course.

Office hours

See the website of Mirko Degli Esposti