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Symplectic Aspects of Weak KAM theory

"The least action principle is one of the most classical tools in the study of convex Hamiltonian systems. It consists in finding specific orbits by minimizing the Lagrangian action functional. Another powerful classical tool in... ver más

31/08/2017

DAUPHINE

840K€

Presupuesto del proyecto: 840K€

Líder del proyecto

UNIVERSITE PARIS DAUPHINE No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2017-08-31 No tenemos la información de la convocatoria

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Líder del proyecto

UNIVERSITE PARIS DAUPHINE No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 840K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto "The least action principle is one of the most classical tools in the study of convex Hamiltonian systems. It consists in finding specific orbits by minimizing the Lagrangian action functional. Another powerful classical tool in Hamiltonian dynamics is the theory of canonical transformations, which provides a large class of admissible changes of coordinates, allowing to put many systems into simplified normal forms. These two tools are difficult to use simultaneously because the Lagrangian action does not behave well under canonical transformations. A large part of the development of symplectic geometry in the second half of the last century consisted in bridging this gap, by developing a framework encompassing a large part of both theories. For example, the direct study of the Hamiltonian action functional (which, as opposed to the Lagrangian action functional, behaves well under canonical transformations) allowed to recover, refine, and generalize beyond the convexity hypothesis, most of the results concerning the existence of periodic orbits which had been proved with the least action principle. Twenty years ago, under the impulsion of John Mather, a renewed use of the least action principle led to the proof of the existence of complicated invariant sets and unstable orbits. This collection of new methods has been called weak KAM theory in view of some similarities with the classical KAM theory. Weak KAM theory, however, uses the least action principle in such a fundamental way that it does not not enter yet into the symplectic framework. My project is to address this problem. This overarching goal federates a number of questions in weak KAM theory, in Hamiltonian dynamics, in symplectic geometry and even in partial differential equations which will be the starting directions of my investigations."

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