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. Hamiltonian formulation of the equations of motion. Flow of a dufferential equation and divergence theorem. Poisson parentheses. Canonical transformations. 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 recommended 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
  

The teacher will also publish on a regular basis, on Virtuale, handwritten notes which cover rather closely (but not exactly!) the material presented in class. Students are invited to use these notes, but only as a supplement to the textbooks.

Teaching methods

Classroom lectures

Assessment methods

The final grade will be the result of a written test and an oral exam. The written test is preliminary to the oral exam. The possible outcomes of the written test are a grade from 16 to 30+, or non-admission to the oral exam. The final grade for the course will be decided at the end of the oral exam, also based on the grade of the written test, but without predetermined constraints: it may be (even substantially) higher or lower than said grade. The mark 30+ indicates a particularly good written test, not the cum laude mention, which is a prerogative of the final grade.

Written test. The written test lasts 3 hours and consists of 3-4 questions, mostly exercises, but with at least one theoretical question.

Oral exam. The oral exam, lasting approximately 30 minutes, is intended to assess the student's knowledge of the various syllabus topics (none excluded!), as well as their ability to logically connect the various parts, appreciating the unity of the subject matter. The ability to express one's thoughts clearly and precisely will also be evaluated.

The grade obtained at a written exam stays valid for the entire corresponding exam period: summer (3 exam sessions), September (1 exam session), or winter (2 exam sessions).

 

Students with specific learning disabilities (SLD) or other temporary or permanent disabilities should contact the University Office for Students with Disabilities and SLD (https://site.unibo.it/studenti-con-disabilita-e-dsa/en) in a timely manner. The office is responsible for proposing any necessary accommodations to students. These accommodations must be submitted to the instructor for approval at least 15 days in advance and will be evaluated based on the course's learning objectives.

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