The theory of Hamiltonian systems originates from the idea that using changes of coordinates mixing positions and speed it should be possible to reduce differential equations to a simple form, thus making possible to understand the behavior of the solutions.
This idea led to the understanding that behind a large class of differential equations (now known as Hamiltonian differential equations) there is a geometric structure which is left invariant (the symplectic structure). The use of symplecti geometry leads to the understanding of many features of the dynamics, in particular of the dynamics of those systems known as integrable systems.
Most Hamiltonian systems, however do not belong to that class and, during the nineteenth and twentieth century some tools were developed in order to understand the dynamics of more general systems.
In the last part of the twentieth century and the first part of this twenty-first century it became clear that many partial differential equations fall into the category of Hamiltonian systems. Some tolls for dealing with such equations were developed.
The course will start with the classical theory. The basic concepts (definition of Hamiltonian systems, canonical transformations, symplectic form) will be introduced. Then the theory of integrable systems will be developed (Liouville Arnold theorem). Such theory will be applied to important special systems (planetary motion, Delauney variables).
The second part of the course will be devoted to the theory of perturbations in which the change in the dynamics of an integrable system under small perturbation will be studied (Birkhoff normal form, KAM theorem). SOme applications will be given (precession of the perihelion of Mercury due to Jupiter, the precession of the equinoxes).
In the last part of the course will be devoted to the basic elements of the theory of Hamiltonian partial differential equations (introduction to Sobolev spaces, the Hamiltonian formulation for PDE, theory of perturbations).
Hamiltonian System 1 mod/02
The laboratory activity will be 'devoted to the numerical study of the dynamics of ordinary differential equations, with particular attention to symplectic algorithms and their theoretical study.