Hamiltonian System 1

The course is divided into three parts: 1) classical theory of finite-dimensional Hamiltonian systems and integrable systems, 2) perturbation theory for finite-dimensional systems, 3) perturbation theory for partial differential equations.

It is also planned to organize some seminars on current topics in scientific research on these subjects.

Part 1. This concerns the theory developed mainly in the nineteenth century and brought to completion by the Arnold-Liouville theorem in the 1960s. This theorem states that if a sufficient number of constants of motion of a Hamiltonian system are known, it is possible to write its solution in a substantially explicit way and, moreover, the flow is equivalent to a linear flow on a torus.

The path toward this theorem is organized into the following points, which will be covered by the course:

1.1 Review of the definition of a Hamiltonian system.

1.2 Characterization of canonical transformations.

1.3 Connection between symplectic geometry and the Hamiltonian formalism.

1.4 The flow of a Hamiltonian system as a canonical transformation, symmetries and conserved quantities.

1.5 Construction of canonical transformations: the generating function.

1.6 Liouville's theorem on the integrability of a system with n degrees of freedom with n independent first integrals in involution.

1.7 Arnold's theorem on equivalence with the linearized flow on a torus, action-angle variables.

1.8 Foliation of phase space into invariant tori.

1.9 Example of the Kepler problem.

Part 2. Perturbation theory and Birkhoff normal form. Since the nineteenth century it has been clear that integrable systems are exceptional and that the vast majority of dynamical systems are not integrable. A typical example is the problem of planetary motion, in which, in addition to the interaction of a planet with the Sun, there is the interaction with all the other planets (which is now known to lead to the existence of chaotic motions). In this part, the main methods for the study of perturbations of integrable systems will be developed.

2.1 The spectrum of linear Hamiltonian systems.

2.2 Perturbation of linear Hamiltonian systems and Birkhoff normal form.

2.2.1 The problem of eliminating nonlinearity and small divisors.

2.2.2 Birkhoff normal form: reduction to normal form and properties of the normal form.

2.2.2 Consequences of the normal form theorem: genericity of the conservation of the energy of harmonic modes.

2.2.3 The resonant case: nonlinear beats.

2.3 Toward KAM theory: perturbation of integrable systems and quasi-persistence of most invariant tori under perturbation.

Part 3. Perturbation theory for partial differential equations and infinite-dimensional systems. Starting in the 1990s, the methods of Hamiltonian mechanics began to be used to study partial differential equations and, on the one hand, to construct periodic solutions or invariant tori in nonlinear equations, and, on the other hand, to analyze the existence time of regular solutions in nonlinear equations. Today there is a fairly satisfactory theory for systems in one spatial dimension, but the case of higher dimension is still largely open.

3.1 The Hamiltonian formalism for partial differential equations.

3.1.1 Two model problems: the wave equation and the Schroedinger equation.

3.1.2 Existence of weak symplectic forms: the case of regular Hamiltonians whose vector field is not even continuous.

3.1.3 Hamiltonian flow and canonicity in the infinite-dimensional case.

3.2 Birkhoff normal form for partial differential equations.

3.2.1 Small divisors of a new type: obstruction to the finite-dimensional procedure.

3.2.2 The Tame condition and the second Melnikov condition.

3.2.3 Birkhoff normal form and almost-global existence in partial differential equations.