The goal of the course is that of acquiring a solid understanding of advanced Hamiltonian mechanics, with particular emphasis on its geometric formulation, and formal aspects of geometric mechanics on Poisson manifolds.
Course Prerequisites
A course in Analytic or Rational Mechanics (in oparticular Lagrangian mechanics and basics of Hamiltonian mechanics). Basics of differential geometry is highly recommended.
Teaching Methods
Lectures. Students will have access to course notes and will be able to interact with each other and with the instructor via the Kiro forum.
Assessment Methods
Oral Exam. The oral exam will consist of a series of questions designed to assess the candidate's level of understanding of the topics covered in the course.
Texts
A. Fasano, S. Marmi “Meccanica Analitica”, Bollati Boringhieri 2002; Notes on Poisson manifolds and Toda systems J. Marsden, T. Ratiu, Introduction to mechanics and symmetry, Springer 1994 Mark Adler, Pierre van Moerbeke, Pol Vanhaecke (auth.) - Algebraic Integrability, Painlevé Geometry and Lie Algebras (2004, Springer)
Contents
Geometric foundations of Lagrangian and Hamiltonian mechanics. Hamiltonian flow, Liouville's theorem, Poincaré's theorem. Symplectic structure of the Hamiltonian phase space; Poincaré-Cartan 1-form and symplectic form. Canonical transformations and their characterization. Algebraic structure of dynamical variables: Poisson brackets and connection with the Lie derivative. Constants of motion and symmetry properties (Hamiltonian Noether's theorem). Hamilton-Jacobi equations; action-angle variables in the one-dimensional case and in the separable n-dimensional case. Completely integrable Hamiltonian systems: Liouville and Arnol'd theorems. Advanced topics for the last part of the course: Poisson geometry, Schouten brackets, singular foliation theory, Liouville integrable systems on Poisson manifolds, Arnold Liouville theorem on Poisson manifolds.
Course Language
Italian
More information
Students in the categories identified by the innovative teaching project will have the opportunity to attend online meetings by appointment at times to be agreed with the teacher, or view the teacher's lecture notes.
