English

At the end of the course, the student should be able to solve problems of analytical mechanics (Hamiltonian formalism, analytical and geometrical methods), symplectic geometry and Poisson geometry.

The student should be able to apply the acquired knowledge to the solution of exercises and relatively basic theoretical problems in the modern approach to mechanics and interable systems.

The student will have acquired a sufficient autonomy to understand which techniques and theoretical background are required to face new problems in above mentioned areas.

The student will acquire the ability to organize a self contained exposition of the subjects he has studied, together with autonomy in finding, reading and understanding textbooks.

First order ODEs and linear ODEs with constant coefficients. Differential

geometry, differentiable manifolds, tensor calculus.

A background in Newtonian and Lagrangian mechanics is required, consisting in the program of the course in analytical mechanics of the Laurea Triennale.

Lagrangian and Hamiltonian systems and their mathematical description in terms of differential geometry.

Lectures on basic and more advanced theoretical topics will be complemented by exercises. Though there is plenty of textbooks, sometimes LaTex typed notes will be distributed. All the exercises will also be distributed, together with their detailed solutions.

Short review of Newtonian mechanics (cardinal equations. Center of Mass and Koenig's theorem).

Short review of Lagrangian Mechanics (Generalized coordinates and tangent vectors.

Lagrange's equations as projection of Newton's equations onto the basis

of tangent vectors. Phase space as tangent bundle. Jacobi energy. Two body problem.

Kepler's problem).

Hamiltonian Mechanics. Hamilton equations on the cotangent bundle

of the configuration space. 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 as canonical

transformations. Noether's theorem. Liuoville and Poincare's theorems. Poisson manifolds, Casimir functions, symplectic foliation. Lie-Poisson brackets and the rigid body.

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.

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B. Dubrovin: Lectures notes "Meccanica analitica" (SISSA).

-- A. Fasano, S. Marmi: {Analytical Mechanics}.

-- L.D. Landau, E.M. Lifsits: {Mechanics}

-- W.M. Boothby: {An Introduction to Differentiable Manifolds and

Riemannian Geometry}.

-- V. Arnold: {Mathematical Methods of Classical Mechanics}.