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Symplectic Aspects of Weak KAM theory

Ziel

"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."

Aufforderung zur Vorschlagseinreichung

ERC-2012-StG_20111012
Andere Projekte für diesen Aufruf anzeigen

Gastgebende Einrichtung

UNIVERSITE PARIS DAUPHINE
EU-Beitrag
€ 840 000,00
Adresse
PLACE DU MARECHAL DE LATTRE DE TASS IGNY
75775 Paris
Frankreich

Auf der Karte ansehen

Region
Ile-de-France Ile-de-France Paris
Aktivitätstyp
Higher or Secondary Education Establishments
Hauptforscher
Patrick Bernard (Prof.)
Kontakt Verwaltung
Edith Buser (Mrs.)
Links
Gesamtkosten
Keine Daten

Begünstigte (1)