Symplectic Homogenization - Laboratoire de Mathématiques d'Orsay Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2023

Symplectic Homogenization

Résumé

Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\gamma$ topology defined by the author, to $\overline{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${\overline H}(y,q,p)$ and thus yields an ``effective Hamiltonian''. We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $\overline H$ coincides with Mather's $\alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard.
Fichier principal
Vignette du fichier
Homogenization-JEP-submitted-2.pdf (777.73 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-00201500 , version 1 (30-12-2007)
hal-00201500 , version 2 (24-02-2023)

Identifiants

Citer

Claude Viterbo. Symplectic Homogenization. 2023. ⟨hal-00201500v2⟩
301 Consultations
78 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More