On the existence of Hamiltonian paths connecting Lagrangian submanifolds

C. R. Math. Rep. Acad. Sci. Canada Vol. 30 (3) 2008, pp. 65–83
(Received: 2008-06-17 )

Nassif Ghoussoub, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2; email: nassif@math.ubc.ca

Abbas Moameni, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2; email: moameni@math.ubc.ca

Abstract/Résumé:

We use a new variational method—based on the theory of anti-selfdual Lagrangians developed recently—to establish the existence of solutions of convex Hamiltonian systems that connect two given Lagrangian submanifolds in \(\mathbb{R}^{2N}\). We also consider the case where the Hamiltonian is only semi-convex. A variational principle is also used to establish existence for the corresponding Cauchy problem.

Une nouvelle méthode variationnelle—basée sur la théorie des Lagrangiens auto-adjoints developée récemment—est utilisée pour établir l’existence de solutions de systèmes Hamiltoniens convexes, qui connectent deux sous-variétés Lagrangiennes données dans \(\mathbb{R}^{2N}\). On considère aussi le cas des Hamiltoniens semi-convexes, ainsi que le problème de Cauchy correspondant.

Keywords: Hamiltonian systems, Lagrangian submanifolds, self-duality

AMS Subject Classification: Hamiltonian structures; symmetries; variational principles; conservation laws 37K05

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