Perturbation theory of hamiltonian systems
A.Y. 2020/2021
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.
Lesson period: First semester (In case of multiple editions, please check the period, as it may vary)
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Course syllabus and organization
Single session
Lesson period
First semester
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.
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.
Prerequisites for admission
Solid knowledge of lagrangian mechanics is advised; differential geometry skills are surely welcome.
Teaching methods
Lectures.
Students are strongly advised to attend the classes.
Students are strongly advised to attend the classes.
Teaching Resources
Lecture notes available on the web page: http://users.mat.unimi.it/users/paleari/didattica/
Assessment methods and Criteria
The final examination consists of an oral exam. 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 examination is passed if the oral part is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
The examination is passed if the oral part is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Professors:
Benedikter Niels Patriz, Paleari Simone
Professor(s)