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Periodic solutions of Hamilton’s equations and local minima of the dual action. (English) Zbl 0596.49030
On considère le système Hamiltonien -p ˙= x H(x,p), x ˙(t)= p H(x,p) pour lequel on cherche des solutions périodiques. On suppose que H est convexe. Moyennant une hypothèse sur le comportement de H au voisinage de (0,0) (resp. au voisinage de l’infini), alors pour tout T assez petit (resp. grand) il existe une solution de période minimale T. La méthode utilisée fait appel au principe d’action duale introduit par l’A. sur lequel il met en évidence des minima locaux.
Reviewer: H.Brezis
MSC:
49S05Variational principles of physics
34C25Periodic solutions of ODE
70H05Hamilton’s equations
49J45Optimal control problems involving semicontinuity and convergence; relaxation
49N15Duality theory (optimization)
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems