Hamiltonian Systems 1

A.Y. 2020/2021
Overall hours
Learning objectives
The main goals of this course are: to provide the basis of Hamiltonian formalism in Classical Mechanics; to provide an introduction to perturbation theory for almost-integrable systems; to illustrate, by means of Lab sessions, some numerical methods for problems arising from Mechanics.
Expected learning outcomes
The student will be able to use the Hamiltonian formalism in the description and analysis of dynamical systems; to apply the main theorems about the dynamics of Hamiltonian systems, or their study; to use perturbation theory techniques in the Hamiltonian case.
Course syllabus and organization

Single session

Lesson period
First semester
The subsequent informations concern the lectures; presently it is not possible to publish the details of the lab part. We suggest to check the web page hosted on Ariel to look for updates.

Video lectures will be made available, hosted on Ariel.
If technically feasible, live lectured will be attempted, probably using Zoom.

Changes in the program and in the references are not expected.

Oral examinations, using Zoom, will be essentially identical to the traditional ones.
Prerequisites for admission
Solid knowledge of lagrangian mechanics is advised; differential geometry skills are surely welcome.
Assessment methods and Criteria
The final examination consists of an oral exam for the first part, and of the evaluation of all the activities performed during the lab sessions for the second module; if such an ongoing evaluation of the lab activities is not feasible, a short project will be possible assigned to the students.

- In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam is based on the activities done in each lab session; in case such evaluation is not feasible, a short project will be assigned to each student.

The examination is passed if the oral part is successfully passed and, for those requiring 9 credits, if the lab activities are positively evaluated. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Hamiltonian System 1 (first part)
Course syllabus
1. Hamiltonian formalism: Hamilton's equations; costants of motion, Poisson brackets; canonical trasformations; Hamilton Jacobi equation.
2. Integrable systems: Liouville's theorem; Arnold-Jost theorem; equilibria in Hamiltonian systems; Symmetries. Lax pairs formalism.
3. Nearly integrable systems: dynamics in a neighbourhood of an elliptic equilibrium; Poincare' theorem; formal perturbative construction of first integrals.
4. Birkhoff normal form. Near the identity canonical trasformations; the Lie series approach, formal expansions and rigorous estimates.
5. Kolmogorov theorem on the peristence of invariant tori supporting quasi periodic motions.
6. Nekhoroshev theorem on exponential stability.
Teaching methods
Students are strongly advised to attend the classes.
Teaching Resources
Lecture notes available on the web page on Ariel.
Hamiltonian System 1 mod/02
Course syllabus
1. Symplectic integrators; general properties and actual code development used to numerically integrate some Hamiltonian system.
2. Explicit perturbative construction of approximated first integral and/or normal forms, by means of symbolic manupulation techniques.
Teaching methods
Lectures and lab classes with computers.
Teaching Resources
Lecture notes available on the web page on Ariel
Hamiltonian System 1 (first part)
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Professor: Paleari Simone
Hamiltonian System 1 mod/02
MAT/07 - MATHEMATICAL PHYSICS - University credits: 3
Laboratories: 24 hours
Lessons: 7 hours
Contact me via email
Office 1039, 1st floor, Dipartimento di Matematica, Via Saldini, 50