Hamiltonian System 1
A.Y. 2025/2026
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
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Course syllabus
The aim of the course is to study some qualitative aspect of the dynamics of finite and infinite dimensional Hamiltonian systems. It is composed of three parts: (1) classical theory and Integrable systems, (2) perturbation theory for finite dimensional systems, (3) perturbation theory for Hamiltonian partial differential equations.
In the first part we will present the elementary tools of Hamiltonian systems and the basic results on the structure of integrable systems. In the second part we will present the main results of perturbation theory for finite dimensional systems. In the third part we will concentrate on modern theory ranging from the beginning of the 2000 to nowadays on perturbation theory for infinite dimensional systems
More in detail
1.1 Hamiltonian formulation of the equations of mechanical systems, canonical transformations, relationship between integrals of motion and symmetries,
1.2 theorem by Liouville-Arnold-Jost on the structure of integrable systems.
1.3 Some explicit examples
2.1 Birkhoff normal form, small divisors, long time stability of the dynamics in perturbation of nonresonant systems, long time behaviour of resonant systems (nonlinear beatings)
2.2 Perturbation theory for nonlinear integrable systems: density of resonances, Poincaré's theorem on non persistence of integrals of motion, Nekhoroshev's theorem on stability over exponentially ling times (Lochak's proof).
Elements of KAM theory. Applications to some important problems including precession of Mercury's perihelion.
3.1 Hamiltonian partial differential equations: the case of the nonlinear wave equation. Birkhoff normal form and almost global existence in 1 space dimension. The case of more than one space dimension.
3.2 Dispersion and decay of local energy in Hamiltonian systems. The example of a linear and nonlinear chain of particles and of the Nonlinear Schroedinger equation.
(Oscillating integrals, Van der Corput lemma and applications)
3.3 Dynamical foundation of statistical mechanics: metastability n the Fermi Pasta Ulam problem.
In the first part we will present the elementary tools of Hamiltonian systems and the basic results on the structure of integrable systems. In the second part we will present the main results of perturbation theory for finite dimensional systems. In the third part we will concentrate on modern theory ranging from the beginning of the 2000 to nowadays on perturbation theory for infinite dimensional systems
More in detail
1.1 Hamiltonian formulation of the equations of mechanical systems, canonical transformations, relationship between integrals of motion and symmetries,
1.2 theorem by Liouville-Arnold-Jost on the structure of integrable systems.
1.3 Some explicit examples
2.1 Birkhoff normal form, small divisors, long time stability of the dynamics in perturbation of nonresonant systems, long time behaviour of resonant systems (nonlinear beatings)
2.2 Perturbation theory for nonlinear integrable systems: density of resonances, Poincaré's theorem on non persistence of integrals of motion, Nekhoroshev's theorem on stability over exponentially ling times (Lochak's proof).
Elements of KAM theory. Applications to some important problems including precession of Mercury's perihelion.
3.1 Hamiltonian partial differential equations: the case of the nonlinear wave equation. Birkhoff normal form and almost global existence in 1 space dimension. The case of more than one space dimension.
3.2 Dispersion and decay of local energy in Hamiltonian systems. The example of a linear and nonlinear chain of particles and of the Nonlinear Schroedinger equation.
(Oscillating integrals, Van der Corput lemma and applications)
3.3 Dynamical foundation of statistical mechanics: metastability n the Fermi Pasta Ulam problem.
Prerequisites for admission
Elements oh Lagrangian Mechanics.
Teaching methods
Lectures and exercises
Teaching Resources
Lectures Notes
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.
- 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.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Laboratories: 12 hours
Lessons: 35 hours
Lessons: 35 hours
Professors:
Bambusi Dario Paolo, Gallone Matteo
Professor(s)