Hamiltonian Systems 1

A.Y. 2019/2020
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

Unique edition

Lesson period
First semester
Prerequisites for admission
Solid knowledge of lagrangian mechanics is advised; differential geometry skills are surely welcome.
Assessement 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.

- 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.

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.
Lecture notes available on the web page: http://users.mat.unimi.it/users/paleari/didattica/
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.
Lecture notes available on the web page: http://users.mat.unimi.it/users/paleari/didattica/
Hamiltonian System 1 (first part)
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Hamiltonian System 1 mod/02
MAT/07 - MATHEMATICAL PHYSICS - University credits: 3
: 24 hours
Lessons: 7 hours