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March 25, 2015 By

On the existence of Hamiltonian paths connecting Lagrangian submanifolds

C. R. Math. Rep. Acad. Sci. Canada Vol. 30 (3) 2008, pp. 65–83

September 30, 2008

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.

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Keywords: Hamiltonian systems, Lagrangian submanifolds, self-duality
AMS Subject Classification: Hamiltonian structures; symmetries; variational principles; conservation laws 37K05

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