Singular Hamiltonian systems and symplectic capacities

Alfred Künzle

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 171-187
  • ISSN: 0137-6934

Abstract

top
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.

How to cite

top

Künzle, Alfred. "Singular Hamiltonian systems and symplectic capacities." Banach Center Publications 33.1 (1996): 171-187. <http://eudml.org/doc/262540>.

@article{Künzle1996,
abstract = {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.},
author = {Künzle, Alfred},
journal = {Banach Center Publications},
keywords = {symplectic capacity; differential inclusion; singular Hamiltonian},
language = {eng},
number = {1},
pages = {171-187},
title = {Singular Hamiltonian systems and symplectic capacities},
url = {http://eudml.org/doc/262540},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Künzle, Alfred
TI - Singular Hamiltonian systems and symplectic capacities
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 171
EP - 187
AB - 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.
LA - eng
KW - symplectic capacity; differential inclusion; singular Hamiltonian
UR - http://eudml.org/doc/262540
ER -

References

top
  1. [A84] J. P. Aubin, L'analyse nonlinéaire et ses motivations économiques, Masson, Paris, 1984. Zbl0551.90001
  2. [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. Zbl0461.49007
  3. [CE80] F. H. Clarke and I. Ekeland, Hamiltonian trajectories with prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103-116. Zbl0403.70016
  4. [Cl81] F. H. Clarke and I. Ekeland, Periodic solutions to Hamiltonian inclusions, J. Differential Equations 40 (1981), 1-6. 
  5. [CW81] B. C. Croke and A. Weinstein, Closed curves on convex hypersurfaces and period of nonlinear oscillations, Invent. Math. 64 (1981), 199-202. Zbl0471.70020
  6. [C77] J. P. Crouzeix, Contribution à l'étude des fonctions quasiconvexes, Thèse de l'Université Clermont-Ferrand II, 1977. 
  7. [C89] J. P. Crouzeix, private communication, 1989. 
  8. [E90] I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer, Berlin, 1990. 
  9. [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. Zbl0641.53035
  10. [EH90] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics II, Math. Z. 203 (1990), 553-567. Zbl0729.53039
  11. [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. Zbl0655.58015
  12. [G85] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. Zbl0592.53025
  13. [HZ90] H.W. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in: Analysis et cetera, Academic Press, 1990, 405-429. 
  14. [K90] A. F. Künzle, Une capacité symplectique pour les ensembles convexes et quelques applications, Ph.D. thesis, Université Paris IX Dauphine, June 1990. 
  15. [K91] A. F. Künzle, The least characteristic action as symplectic capacity, preprint, Forschungsinstitut für Mathematik, ETH Zürich, May 1991. 
  16. [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. 
  17. [K95] A. F. Künzle, On the symplectic capacities extending the least characteristic action of convex sets, preprint EPFL, August 1995. 
  18. [K95a] A. F. Künzle, The symplectic product of hypersurfaces, in preparation. 
  19. [P99] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, tome III, Paris, 1899. 
  20. [R70] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970. 
  21. [R81] R. T. Rockafellar, The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions, Heldermann, Berlin, 1981. 
  22. [R89] R. T. Rockafellar, manuscript, Varenna, June 1989. 
  23. [Si90] J. C. Sikorav, Systèmes Hamiltoniens et topologie symplectique, Lecture notes, Dipartimento di Matematica dell'Università di Pisa, August 1990. 
  24. [V87] C. Viterbo, A proof of Weinstein’s conjecture in ℝ 2 n , Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 337-357. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.