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Banach Center Publications

1996 | 33 | 1 | 171-187
Tytuł artykułu

Singular Hamiltonian systems and symplectic capacities

Autorzy
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
Czasopismo
Rocznik
Tom
Numer
Strony
171-187
Opis fizyczny
Daty
wydano
1996
Twórcy
autor
• Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany
Bibliografia
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• [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.
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• [G85] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
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• [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.
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• [V87] C. Viterbo, A proof of Weinstein's conjecture in $ℝ^2n$, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 337-357.
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Bibliografia
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