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.
@article{bwmeta1.element.bwnjournal-article-bcpv33z1p171bwm,
author = {K\"unzle, Alfred},
title = {Singular Hamiltonian systems and symplectic capacities},
journal = {Banach Center Publications},
volume = {37},
year = {1996},
pages = {171-187},
zbl = {0855.58027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv33z1p171bwm}
}
Künzle, Alfred. Singular Hamiltonian systems and symplectic capacities. Banach Center Publications, Tome 37 (1996) pp. 171-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv33z1p171bwm/
[000] [A84] J. P. Aubin, L'analyse nonlinéaire et ses motivations économiques, Masson, Paris, 1984. | Zbl 0551.90001
[001] [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. | Zbl 0461.49007
[002] [CE80] F. H. Clarke and I. Ekeland, Hamiltonian trajectories with prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103-116. | Zbl 0403.70016
[003] [Cl81] F. H. Clarke and I. Ekeland, Periodic solutions to Hamiltonian inclusions, J. Differential Equations 40 (1981), 1-6.
[004] [CW81] B. C. Croke and A. Weinstein, Closed curves on convex hypersurfaces and period of nonlinear oscillations, Invent. Math. 64 (1981), 199-202. | Zbl 0471.70020
[005] [C77] J. P. Crouzeix, Contribution à l'étude des fonctions quasiconvexes, Thèse de l'Université Clermont-Ferrand II, 1977.
[006] [C89] J. P. Crouzeix, private communication, 1989.
[007] [E90] I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer, Berlin, 1990.
[008] [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. | Zbl 0641.53035
[009] [EH90] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics II, Math. Z. 203 (1990), 553-567. | Zbl 0729.53039
[010] [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. | Zbl 0655.58015
[011] [G85] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. | Zbl 0592.53025
[012] [HZ90] H.W. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in: Analysis et cetera, Academic Press, 1990, 405-429.
[013] [K90] A. F. Künzle, Une capacité symplectique pour les ensembles convexes et quelques applications, Ph.D. thesis, Université Paris IX Dauphine, June 1990.
[014] [K91] A. F. Künzle, The least characteristic action as symplectic capacity, preprint, Forschungsinstitut für Mathematik, ETH Zürich, May 1991.
[015] [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.
[016] [K95] A. F. Künzle, On the symplectic capacities extending the least characteristic action of convex sets, preprint EPFL, August 1995.
[017] [K95a] A. F. Künzle, The symplectic product of hypersurfaces, in preparation.
[018] [P99] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, tome III, Paris, 1899.
[019] [R70] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.
[020] [R81] R. T. Rockafellar, The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions, Heldermann, Berlin, 1981.
[021] [R89] R. T. Rockafellar, manuscript, Varenna, June 1989.
[022] [Si90] J. C. Sikorav, Systèmes Hamiltoniens et topologie symplectique, Lecture notes, Dipartimento di Matematica dell'Università di Pisa, August 1990.
[023] [V87] C. Viterbo, A proof of Weinstein’s conjecture in , Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 337-357.