Hamiltonian systems as selfdual equations

Abstract

Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative action functionals obtained by a generalization of Bogomolnyi’s trick of ‘completing squares’. Reminiscent of the selfdual Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the corresponding Euler-Lagrange equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study certain Lagrangian intersections.

References

1. Aubin J P Ekeland I Applied Nonlinear AnalysisMineolaDoverPublications, Inc 2006
2. Brezis H Ekeland I Un principe variationnel associé ècertaines equations paraboliques. Le cas independant du tempsC R Acad Sci Paris Sér A 1976 282971974
3. Ekeland I ConvexityMethods in Hamiltonian MechanicsBerlinSpringer-Verlag 1990
4. Ekeland I Temam R Convex Analysis and Variationalproblems. Classics in Applied Mathematics, 28PhiladelphiaSIAM 1999
5. Fan Ky MinimaxtheoremsProc Nat Acad Sci U S A 1953 394247. doi:10.1073/pnas.39.1.42
6. Ghoussoub N Anti-selfdualLagrangians: Variational resolutions of non self-adjoint equationsand dissipative evolutionsAIHP-AnalyseNon Linéaire 2007 24171205
7. Ghoussoub N Anti-symmetricHamiltonians: Variational resolution of Navier-Stokes equations andother nonlinear evolutionsComm Pure &Applied Math 2007 60(5)619653. doi:10.1002/cpa.20176
8. Ghoussoub N SelfdualPartial Differential Systems and their Variational Principles. Universitext.BerlineSpringer-Verlag 2007
9. Ghoussoub N Moameni A On the existence of Hamiltonianpaths connecting Lagrangian submanifolds 2005 112
10. Ghoussoub N Moameni A Selfdual variational principlesfor periodic solutions of Hamiltonian and other dynamical systemsComm in PDE 2007 32771795
11. Ghoussoub N Moameni A Hamiltonian systems of PDEswith selfdual boundary conditions 2007 136
12. Mawhin J Willem M Critical Point Theory and HamiltonianSystemsApplied Mathematical Sciences, 74BerlinSpringer-Verlag 1989