Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

Duke Mathematical Journal

Volumn 144, Issue 2, 2008, Pages 235-284

  Cardin, Franco ^a   Viterbo, Claude ^b  

^a UNIVERSITY OF PADOVA   (Italy)

^b CENTRE DE MATHÉMATIQUES LAURENT SCHWARTZ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 50349096877     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/00127094-2008-036     Document Type: Article

References (46)

1. G. BARLES, Solutions de viscosité des équations de Hamilton-Jacobi, Math. Appl. (Berlin) 17, Springer, Paris, 1994. MR 1613876
2. G. BARLES and A. TOURIN, Commutation properties of semigroups for first-order Hamilton-Jacobi equations and application to multi-time equations, Indiana Univ. Math. J. 50 (2001), 1523-1544. MR 1889069
3. S. BENENTI, "Symplectic relations in analytical mechanics" in Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Vol. I (Torino, Italy, 1982), Atti Accad. Sci. Torino C1. Sci. Fis. Mat. Natur. 117, suppl. 1, Acad. Sci. Torino, Turin, Italy, 1983, 39-91. MR 0773479
4. S. BENENTI and W. M. TULCZYJEW, "The geometrical meaning and globalization of the Hamilton-Jacobi method" in Differential Geometrical Methods in Mathematical Physics (Aix-en-Provence, France; Salamanca, Spain, 1979), Lecture Notes in Math. 836, Springer, Berlin, 1980, 9-21. MR 0607684
5. O. BERNARDI and F. CARDIN, Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case, Commun. Pure Appl. Anal. 5 (2006), 793-812. MR 2246008
6. A. BOREL, Cohomologie des espaces localement compacts d'après J. Leray, 3rd ed., Lecture Notes in Math. 2, Springer, Berlin, 1964. MR 0174045
7. M. BRUNELLA, On a theorem of Sikorav, Enseign. Math. (2) 37 (1991), 83-87. MR 1115746
8. M. CHAPERON, Une idée du type "géodésiques brisées" pour les systèmes hamiltoniens, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), 293-296. MR 0765426
9. _, Lois de conservation et géométrie symplectique, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 345-348. MR 1094198
10. Y. V. CHEKANOV, Critical points of quasifunctions, and generating families of Legendrian manifolds (in Russian), Funktsional. Anal. i Prilozhen. 30, no. 2 (1996), 56-69;
11. translation in Funct. Anal. Appl. 30 (1996), 118-128. MR 1402081
12. M. G. CRANDALL and P.-L. LIONS, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277, no. 1 (1983), 1-42. MR 0690039
13. B. DACOROGNA and P. MARCELLINI, Implicit Partial Differential Equations, Progr. Nonlinear Differential Equations Appl. 37, Birkhäuser, Boston, 1999. MR 1702252
14. I. EKELAND and H. HOFER, Ṡymplectic topology and Hamiltonian dynamics, Math. Z. 200 (1989), 355-378. MR 0978597
15. Y. ELIASHBERG, Rigidity of symplectic and contact structures, preprint, 1981.
16. M. ENTOV, L. POLTEROVICH, and F. ZAPOLSKY, Quasi-morphisms and the Poisson bracket, preprint, arXiv:math/0605406v2 [math.SG]
17. A. FATHI and E. MADERNA, Weak KAM theorem on non compact manifolds, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1-27. MR 2346451
18. M. GROMOV, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307 - 347. MR 0809718
19. V. GUILLEMIN and A. POLLACK, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J., 1974. MR 0348781
20. H. HOFER, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 25-38. MR 1059642
21. V. HUMILIÈRE, On some completions of the space of Hamiltonian maps, preprint, arXiv:math/0511418v3 [math.SG]
22. 0 closed, in preparation, 2006.
23. F. LAUDENBACH and J.-C. SIKORAV, Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibré cotangent, Invent. Math. 82 (1985), 349-357. MR 0809719
24. _, Hamiltonian disjunction and limits of Lagrangian submanifolds, Internat. Math. Res. Notices 1994, no. 4, 161 - 168. MR 1266111
25. A. LICHNEROWICZ, Variété symplectique et dynamique associée à une sous-variété, C. R. Acad. Sci. Paris Sér. A-B 280(1975), A523-A527. MR 0405503
26. P.-L. LIONS and J.-C. ROCHET, Hopf formula and multitime Hamilton-Jacobi equations, Proc. Amer. Math. Soc. 96 (1986), 79-84. MR 0813815
27. M. MOTTA and F. RAMPAZZO, Nonsmooth multi-time Hamilton-Jacobi systems, Indiana Univ. Math. J. 55 (2006), 1573-1614. MR 2270930
28. M. R. MENZIO and W. M. TULCZYJEW, Infinitesimal symplectic relations and generalized Hamiltonian dynamics, Ann. Inst. H. Poincaré Sect. A (N.S.) 28 (1978), 349-367. MR 0511066
29. F. RAMPAZZO and H. SUSSMANN, "Set-valued differentials and a nonsmooth version of Chow's theorem" in Proceedings of the 40th IEEE Conference on Decision and Control, Vol. 3 (Orlando, Fla., 2001), IEEE, New York, 2001, 2613-2618.
30. J.-C. ROCHET, The taxation principle and multitime Hamilton-Jacobi equations, J. Math. Econom. 14 (1985), 113-128. MR 0827147
31. E. SCHRÖDINGER, Quantisierung als Eigenwertproblem, II, Ann. Physik (4) 79 (1926), 489-527.
32. J.-C. SIKORAV, Problèmes d'intersections et de points fixes en géométrie hamiltonienne, Comment. Math. Helv. 62 (1987), 62-73. MR 0882965
33. _, "Quelques propriétés des plongements lagrangiens" in Analyse globale et physique mathématique (Lyon, 1989), Mém. Soc. France (N.S.) 46, Soc. Math. France, Montrouge, 1991, 151-167. MR 1125841
34. _, personal communication, 1990.
35. D. THÉRET, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology Appl. 96 (1999), 249-266. MR 1709692
36. _, Utilisation des fonctions génératrices en géométrie symplectique global, Ph.D. dissertation, Université Paris Diderot-Paris 7, Paris, 1996.
37. W. M. TULCZYJEW, Relations symplectiques et les équations d'Hamilton-Jacobi relativistes, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), A545-A547. MR 0467826
38. C. VITERBO, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685-710. MR 1157321
39. _, "Solutions of Hamilton-Jacobi equations and symplectic geometry: Addendum to: Séminaire sur les Équations an Dérivées Partielles, 1994-1995 (Palaiseau, France, 1995)" in Séminaire sur les Équations aux Dérivées Partielles, 1995-1996 (Palaiseau, France, 1995), Semin. Equ. Deriv. Partielles, École Polytech., Palaiseau, France, 1996, MR 1604386
40. _, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc. 13 (2000), 411-431. MR 1750956
41. _, On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows, Int. Math. Res. Not. 2006, no. 34028;
42. Int. Math. Res. Not. 2006, no. 38784. MR 2233715; MR 2272093
43. C. VITERBO and A. OTTOLENGHI, Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique, preprint, 1995.
44. A. WEINSTEIN, Lectures on Symplectic Manifolds, CBMS Regional Conf. Ser. in Math. 29, Amer. Math. Soc., Providence, 1979. MR 0598470
45. T. ZHUKOVSKAYA, Metamorphoses of the Chaperon-Sikorav weak solutions of Hamilton-Jacobi equations, J. Math. Sci. 82 (1996), 3737-3746. MR 1428734
46. _, Singularités de minimax et solutions faibles d'Equations aux dérivées partielles, Ph.D. dissertation, Université Paris Diderot-Paris 7, Paris, 1993.