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Contenuto archiviato il 2024-06-18

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 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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Meccanismo di finanziamento

ERC-SG - ERC Starting Grant

Istituzione ospitante

Contributo UE
€ 840 000,00
75775 Paris

Mostra sulla mappa

Ile-de-France Ile-de-France Paris
Tipo di attività
Higher or Secondary Education Establishments
Ricercatore principale
Patrick Bernard (Prof.)
Contatto amministrativo
Edith Buser (Mrs.)
Costo totale
Nessun dato

Beneficiari (1)