00686 - Analytical Mechanics (M-Z)

Learning outcomes

By the end of the course, students will have solid knowledge of Lagrangian and Hamiltonian mechanics and of the main integrable models. The learning outcomes are structured across three levels of depth, in accordance with the Dublin Descriptors and EQF Level 6.

1. Knowledge and Understanding (DD 1)

• Know the foundations of Lagrangian and Hamiltonian formalism and their variational basis.
• Understand the physical and geometric meaning of the main structures of analytical mechanics (phase space, manifolds, symmetries).
• Know the fundamental theorems: Noether, Liouville, stability of equilibria.

2. Applying Knowledge and Understanding (DD 2)

• Write the Lagrangian and Hamiltonian functions of a given mechanical system, applying variational principles.
• Analyse the phase space of one-dimensional systems, identifying equilibria and their stability.
• Recognise first integrals of motion related to symmetries and apply Noether's theorem to concrete cases.
• Solve the equations of motion in the small oscillation approximation (normal frequencies, parametric resonance).
• Analytically discuss the solutions for the central force problem (Kepler's laws) and for the rigid body (spinning top).
• Apply elementary perturbative methods to the study of mechanical systems.

3. Making Judgements, Communication and Learning Skills (DD 3-4-5)

• Critically assess the formal consistency of a mathematical-physical argument, distinguishing rigorous proofs from qualitative reasoning.
• Express mathematical arguments orally and in writing with precision and rigour, using the language appropriate to analytical mechanics.
• Independently read advanced scientific texts (such as Arnold), identifying their logical structure and main results.
• Develop modelling skills: translate a physical system into a mathematical model, selecting the most appropriate coordinates and formalism.

Course contents

The course content is organized into core topics (mandatory for the exam) and advanced topics (valuable for your formation, but not subject to detailed assessment). This distinction is communicated explicitly throughout the course.

Core topics

• Principles of Relativity and Determinism: the Galilean group and Newton's equations.
• Equations of motion: systems with one and two degrees of freedom.
• Conservative force fields and angular momentum.
• Motion in a central force field: Kepler's laws.
• Variational principles: Lagrange's and Hamilton's equations.
• The Legendre transformation.
• Lagrangian mechanics on manifolds: holonomic constraints and Lagrangian dynamical systems.
• Noether's theorem.
• D'Alembert's principle.
• Small oscillations: normal frequencies and parametric resonance.
• Rigid body dynamics: composition of motions, inertial forces, Coriolis force, Euler's equations.

Advanced topics

• Liouville's theorem and its geometric implications.
• The full canonical formalism.
• Integrable systems: theory and classification.

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

The course consists of traditional frontal lectures (blackboard and slides), complemented by weekly structured exercise sessions.

Assessment methods

Written and Oral Evidence

Teaching tools

The written exam lasts 3 hours and consists of 3–4 questions: mostly applied exercises, but including at least one theoretical question. The possible outcomes are a grade from 16 to 30+, or rejection from the oral exam. The outcome 30+ indicates a particularly strong written performance and does not correspond to honours (cum laude), which is reserved for the final grade.

A grade obtained in a written exam remains valid for the entire corresponding exam session: summer (3 sittings), September (1 sitting), or winter (2 sittings).

Oral exam

The oral exam, lasting approximately 30 minutes, is designed to assess:

• knowledge of all topics covered in the course (none excluded);
• the ability to logically connect the various topics, appreciating the unity of the subject;
• the ability to express one's reasoning clearly and precisely.

The final grade is decided at the conclusion of the oral exam, taking into account the written exam grade as well, but without any predetermined constraints: it may be (even substantially) higher or lower than the written grade.

Office hours

See the website of Mirko Degli Esposti