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Counting Reeb Chords on spherizations
Auteur(s)
Wullschleger, Raphael Elias
Editeur(s)
Date de parution
2014
Résumé
In classical physics, one is interested in finding solutions of the Newtonian equations of motion. If there is a certain number of bodies which attract each others and if one assumes an initial configuration of these masses, then one would like to understand the time evolution of this system according to Newton’s equations, i.e. the change of position and momentum of all these bodies as functions of time. But already in the case of three bodies – say, the moon, the sun and the earth – one knows only very little and this question remains essentially unanswered. <br> Rewriting the Newtonian equations of motion in an equivalent way leads to Hamilton’s equations. Solutions of Hamilton’s equations are paths – in physical terms – in phase space, whereas in mathematical terms one calls this space the cotangent bundle. So classical physical evolution takes mathematically place in cotangent bundles. <br> Symplectic geometry is a new and prominent subject within differential geometry, one of the few basic branches of mathematics. The cotangent bundle is probably the most famous representative of a so-called symplectic manifold. It holds true that the old physical questions got via the steps explained above a new and strong geometrical interpretation. <br> Floer homology is a powerful tool to study solutions of Hamilton’s equations. It gives the possibility to use topological information about the cotangent bundle to obtain qualitative and quantitative results on solutions of Hamilton’s equations. <br> The energy is a property of a physical system which remains constant during evolution of time. Therefore, it is natural to look for solutions of Hamiltonian systems on surfaces as certain subsets – called energy hyper-surfaces – of cotangent bundles which are characterized by the fact that the energy function takes for all points of these surfaces the same value. Roughly speaking, solutions of Hamilton’s equations along energy hypersurfaces are called Reeb chords. The spectrum of such an energy hypersurface is simply the set of all times needed to move along the paths which are solutions of Hamilton’s equations. So it is the set of times required to walk along the Reeb chords of a given energy hypersurface. In particular, the counting function associated to an energy hypersurface is studied. This function calculates the number of solutions whose times are shorter than a given value. <br> In this thesis, steps are taken towards an understanding of the time spectrum of fiberwise starshaped hypersurfaces in cotangent bundles. The base manifold is throughout assumed to be a closed connected Riemannian manifold. It is shown that under the additional assumption of exponential-resp. polynomial growth of the fundamental group of the base manifold, the counting function grows at least exponentially resp. at least polynomially in time. Generally, for every fiberwise starshaped hypersurface over a closed connected Riemannian manifold, the associated counting function grows at least linearly in time. These are asymptotic results. Afterwards the question of understanding fast Reeb chords is considered. An estimate for the time of the fastest resp. of the second fastest Reeb chord is given. More specifically, this question is addressed by choosing special base manifolds, or configuration spaces, such as Lie groups or generally (Riemannian) symmetric spaces. Estimates for the times of the <i>k</i>fastest Reeb chords are deduced. These estimates depend on the geometry of the base manifold only. Another attempt is of group theoretic nature. If the fundamental group of the base manifold is of order <i>k</i>, then there are at least <i>k</i> Reeb chords satisfying an upper time bound <i>k</i> times the diameter of the (compact) base manifold. Finally, some results concerning the stability of the time of the fastest Reeb chord are presented.
Notes
Keywords: Hamiltonian dynamics ; Symplectic geometry ; Lagrangian Floer homology ; contact geometry ; Reeb dynamics Thèse de doctorat : Université de Neuchâtel, 2014
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Type de publication
doctoral thesis
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