At the end of the course, the student should be able to solve problems of
analytical mechanics, symplectic geometry and Poisson geometry.

The student will have acquired a sufficient autonomy to understand
which techniques and theoretical background are required to face new
problems in the areas of the course. The student will acquire autonomy in
finding, reading and understanding textbooks.

First order ODEs and linear ODEs with constant coefficients. Differentiable
manifolds, tensor calculus.

A background in Newtonian and Lagrangian mechanics as covered by the
program of the course in analytical mechanics of the Laurea Triennale.

Lagrangian and Hamiltonian systems and their mathematical description
using differential geometry.

-- A. Fasano, S. Marmi: (Analytical Mechanics}.
-- W.M. Boothby: {An Introduction to Differentiable Manifolds and
Riemannian Geometry}.
-- Ana Cannas da Silva (Lectures on symplectic Geometry).
-- V. Arnold: {Mathematical Methods of Classical Mechanics}.

Short review of Lagrangian Mechanics (Generalized coordinates and
tangent vectors. Euler-Lagrange equations. Phase space as tangent
bundle. Noether's theorem). Kepler's problem). Geodesic motions.
Hamiltonian Mechanics. Hamilton equations on the cotangent bundle.
Canonical transformations and generating functions. Hamilton-Jacobi
equation and its integration. Liouville's theorem and completely
integrable systems. Arnold's theorem, action-angle variables.
Poisson brackets on the cotangent bundle and canonical transformations.
Symplectic forms and manifolds, Darboux' theorem. Hamiltonian vector
fields and Hamiltonian flows. Liuoville and Poincare's theorems.
Poisson manifolds, Casimir functions, symplectic foliation. Lie-Poisson
brackets and the rigid body

Lectures on basic and more advanced theoretical topics will be
complemented by exercises.

Written and oral exams. The written exam consists of exercises similar to
those given at classes. The oral exam is based on questions about the
program of the course.