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1996 | 33 | 1 | 171-187
Tytuł artykułu

Singular Hamiltonian systems and symplectic capacities

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The purpose of this paper is to develop the basics of a theory of Hamiltonian systems with non-differentiable Hamilton functions which have become important in symplectic topology. A characteristic differential inclusion is introduced and its equivalence to Hamiltonian inclusions for certain convex Hamiltonians is established. We give two counterexamples showing that basic properties of smooth systems are violated for non-smooth quasiconvex submersions, e.g. even the energy conservation which nevertheless holds for convex submersions. This also implies that the convexity assumption determines, although not symplectically invariant, a limit case for symplectic geometry. Some applications of this theory are reviewed: symplectic capacities for general convex sets, the symplectic product and a product formula for symplectic capacities.
Słowa kluczowe
Rocznik
Tom
33
Numer
1
Strony
171-187
Opis fizyczny
Daty
wydano
1996
Twórcy
  • Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany
Bibliografia
  • [A84] J. P. Aubin, L'analyse nonlinéaire et ses motivations économiques, Masson, Paris, 1984.
  • [Cl80] F. H. Clarke, The Erdmann condition and Hamiltonian inclusions in optimal control and the calculus of variations, Canad. J. Math. 32 (1980), 494-509.
  • [CE80] F. H. Clarke and I. Ekeland, Hamiltonian trajectories with prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103-116.
  • [Cl81] F. H. Clarke and I. Ekeland, Periodic solutions to Hamiltonian inclusions, J. Differential Equations 40 (1981), 1-6.
  • [CW81] B. C. Croke and A. Weinstein, Closed curves on convex hypersurfaces and period of nonlinear oscillations, Invent. Math. 64 (1981), 199-202.
  • [C77] J. P. Crouzeix, Contribution à l'étude des fonctions quasiconvexes, Thèse de l'Université Clermont-Ferrand II, 1977.
  • [C89] J. P. Crouzeix, private communication, 1989.
  • [E90] I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer, Berlin, 1990.
  • [EH89] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics I, Math. Z. 200 (1989) 355-378; see also C. R. Acad. Sci. Paris Sér. I 307 (1988), 37-40.
  • [EH90] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics II, Math. Z. 203 (1990), 553-567.
  • [El87] Y. Eliashberg, A theorem on the structure of wave fronts and its applications to symplectic topology, Functional. Anal. and Appl. 21 (1987), 227-232.
  • [G85] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
  • [HZ90] H.W. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in: Analysis et cetera, Academic Press, 1990, 405-429.
  • [K90] A. F. Künzle, Une capacité symplectique pour les ensembles convexes et quelques applications, Ph.D. thesis, Université Paris IX Dauphine, June 1990.
  • [K91] A. F. Künzle, The least characteristic action as symplectic capacity, preprint, Forschungsinstitut für Mathematik, ETH Zürich, May 1991.
  • [K94] A. F. Künzle, On the minimal number of closed characteristics on hypersurfaces diffeomorphic to a sphere, preprint, Max-Planck-Institut, Bonn, no. 94-4.
  • [K95] A. F. Künzle, On the symplectic capacities extending the least characteristic action of convex sets, preprint EPFL, August 1995.
  • [K95a] A. F. Künzle, The symplectic product of hypersurfaces, in preparation.
  • [P99] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, tome III, Paris, 1899.
  • [R70] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.
  • [R81] R. T. Rockafellar, The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions, Heldermann, Berlin, 1981.
  • [R89] R. T. Rockafellar, manuscript, Varenna, June 1989.
  • [Si90] J. C. Sikorav, Systèmes Hamiltoniens et topologie symplectique, Lecture notes, Dipartimento di Matematica dell'Università di Pisa, August 1990.
  • [V87] C. Viterbo, A proof of Weinstein's conjecture in $ℝ^2n$, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 337-357.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv33z1p171bwm
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