00686 - Analytical Mechanics (A-L)

Learning outcomes

At the end of the course, the student possesses a basic knowledge of Lagrangian and Hamiltonian mechanics and their integrable models. In particular, the student is able to write the Lagrangian and Hamiltonian functions of a mechanical system; describe the phase space of one-dimensional systems; determine the existence of constants of motion related to symmetries; study the stability of equilibria and find the laws of motion in the approximation of small oscillations; describe the laws of motion for central fields and spinning tops; use variational principles to write the equations of motion and apply perturbative methods.

Course contents

Tentative syllabus:

Recap of Newtonian mechanics: one-particle and many-particle systems, conservation principles. Integration of motion: systems with one degree of freedom. Central fields. Kepler's problem. Lagrangian formulation of the equations of motion. Hamilton's Least Action Principle. Basic notions of Differential Geometry. Lagrangian mechanics of constrained systems. D'Alembert's Principle. Principle of Least Constrained Action. Noether's Theorem. Legendre transform. Hamiltonian formulation of the equations of motion. Small oscillations.  Mobile systems of coordinate: relative kinematics and dynamics. The rigid body.

Readings/Bibliography

The reference textbook for the course is:

• V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag

This is a rather advanced textbook, which will be covered at a suitable pace for students of this class, simplifying and integrating the various topics. Other reference textbooks (in order of importance for this course) are: 

• H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd ed., Pearson
  
• L. D. Landau, E. M. Lifshits, Mechanics (Course of Theoretical Physics Volume 1), 3rd ed., Butterworth-Heinemann
  

Teaching methods

Classroom lectures

Assessment methods

Written and oral exams

Teaching tools

Teacher's notes and printed companion notes of Arnold's textbook by Profs. Degli Esposti, Graffi, Isola.

Office hours

See the website of Marco Lenci