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Complexity and recurrence in Hamiltonian systems

Titel Englisch Complexity and recurrence in Hamiltonian systems
Gesuchsteller/in Schlenk Felix
Nummer 125352
Förderungsinstrument Projektförderung (Abt. I-III)
Forschungseinrichtung Institut de mathématiques Université de Neuchâtel
Hochschule Universität Neuenburg - NE
Hauptdisziplin Mathematik
Beginn/Ende 01.04.2009 - 30.09.2010
Bewilligter Betrag 167'980.00
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Keywords (4)

Hamiltonian Dynamics; Lagrangian submanifolds; Lagrangian tori; symplectic rigidity

Lay Summary (Englisch)

Lead
Lay summary
Hamiltonian systems are dynamical systems that describe "systems without frictions". Since in such systems small oscillations never decay, such system are often complicated, and interesting.Typical examples of Hamiltonian systems are the dynamical systems of classical mechanics, such as the pendulum, the motion of a particle in a force field, and the motion of stars and planets.Many features of Hamiltonian systems can be analyzed and better understood by studying the geometry of the symplectic manifold on which these systems live.Lagrangian submanifolds are submanifolds of half the dimension on which the symplectic form vanishes.After a saying of A. Weinstein, "everything is Lagrangian''. By this he in particular meant that "all'' concepts about Hamiltonian systems can be translated to properties of Lagrangian submanifolds in symplectic manifolds.In particular, symplectic rigidity(i.e., properties of Hamiltonian systems that do not hold for all smooth or volume preserving flows) can often be rephrased in terms of (non-)existence results for Lagrangian submanifolds and in terms of their intersections under Hamiltonian deformations. Particularly interesting Lagrangian submanifolds are the so-called monotone ones, that are in a symplectic sense symmetric. Lagrangian tori are interesting because they are particularly natural in dynamics, in view of the Arnold--Liouville--Jost theorem and in view of KAM theory.Already in such simple symplectic manifolds as standard symplectic space~$R^{2n}$ or the product of $2$-spheres or complex projective space endowed with the standard Kähler form, not many monotone Lagrangian tori were known.In this ongoing project, we construct in these spaces many monotone Lagrangian tori.
Direktlink auf Lay Summary Letzte Aktualisierung: 21.02.2013

Verantw. Gesuchsteller/in und weitere Gesuchstellende

Mitarbeitende

Verbundene Projekte

Nummer Titel Start Förderungsinstrument
144432 From symplectic embeddings to number theory II 01.10.2012 Projektförderung (Abt. I-III)
132000 From symplectic embeddings to number theory 01.10.2010 Projektförderung (Abt. I-III)

Abstract

The most interesting dynamical systems are those describing systems without friction, and many among these can be described as Hamiltonian systems. Since there is no friction, one can expect that Hamiltonian systems are complicated. This is what we aim to prove in the first part of the project for many natural Hamiltonian systems.The origin of Hamiltonian dynamics is the study of the motion of planets, which move - fortunately! - on (almost) closed orbits. The search for closed orbits is therefore a fundamental and particularly beautiful problem in this field. In the second part of the project, we aim to show that for our class of Hamiltonian systems, every energy surface carries infinitely many closed orbits.We shall first try to prove this for a special but very natural class of Hamiltonian systems.The configuration spaces we consider are those closed manifolds whose (based or free) loop space is complicated in the sense that the dimension of its homology grows exponentially fast.Most closed manifolds belong to this class.The idea is that the fast homological growth of the configuration space Q should lead to a fast growth of the orbit complexity and of the number of closed orbits for ALL natural Hamiltonian flows over Q at each energy level.In technical terms, the first project aims to show that on rationally hyperbolic manifolds, each Reeb flow on the spherization has positive topological entropy.This would extend famous results by Dinaburg, Gromov and Paternain on geodesic flows.The second project aims to show that the number of closed orbits of these Reeb flows grows exponentially fast in time. This would extend work by Gromov on closed geodesics, and would be a strong quantitative version of the Weinstein conjecture for these systems.
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