×

zbMATH — the first resource for mathematics

Symplectic geometry. (English) Zbl 0465.58013

MSC:
53D50 Geometric quantization
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
81Q99 General mathematical topics and methods in quantum theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70Hxx Hamiltonian and Lagrangian mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. · Zbl 0393.70001
[2] V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. V. I. Arnol\(^{\prime}\)d, Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60.
[3] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255 – 354. · Zbl 0233.22005 · doi:10.1007/BF01389744 · doi.org
[4] G. D. Birkhoff, Fifty years of American mathematics, Semicentennial Addresses of Amer. Math. Soc., 1938, p. 307. · JFM 64.0003.04
[5] C. E. Bond, Biology of fishes, Sanders, Philadelphia, Pa., 1979.
[6] L. Boutet de Monvel, private communication.
[7] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. · Zbl 0469.47021
[8] Shiing Shen Chern, From triangles to manifolds, Amer. Math. Monthly 86 (1979), no. 5, 339 – 349. · Zbl 0425.53002 · doi:10.2307/2321093 · doi.org
[9] J. J. Duistermaat, Applications of Fourier integral operators, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 263 – 268. · Zbl 0341.35072
[10] J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), no. 3-4, 183 – 269. · Zbl 0232.47055 · doi:10.1007/BF02392165 · doi.org
[11] V. E. Zaharov and L. D. Faddeev, The Korteweg-de Vries equation is a fully integrable Hamiltonian system, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 18 – 27 (Russian).
[12] Victor Guillemin, Clean intersection theory and Fourier integrals, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974) Springer, Berlin, 1975, pp. 23 – 35. Lecture Notes in Math., Vol. 459.
[13] V. W. Guillemin and S. Sternberg, Geometric asymptotics, Amer. Math. Soc., Providence, R. I., 1976.
[14] Victor Guillemin and Shlomo Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), no. 4, 915 – 955. · Zbl 0446.58019 · doi:10.2307/2373923 · doi.org
[15] P. de la Harpe and M. Karoubi, Perturbations compactes des representations d’un groupe dans un espace de Hilbert, Bull. Soc. Math. France Mém. 46 (1976), 41-65. · Zbl 0331.46051
[16] Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79 – 183. · Zbl 0212.46601 · doi:10.1007/BF02392052 · doi.org
[17] D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), no. 4, 481 – 507. · Zbl 0368.58008 · doi:10.1002/cpa.3160310405 · doi.org
[18] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57 – 110 (Russian). · Zbl 0090.09802
[19] A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt; Grundlehren der Mathematischen Wissenschaften, Band 220. · Zbl 0342.22001
[20] Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 87 – 208. Lecture Notes in Math., Vol. 170. · Zbl 0223.53028
[21] Bertram Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), no. 3, 195 – 338. · Zbl 0433.22008 · doi:10.1016/0001-8708(79)90057-4 · doi.org
[22] J. L. Lagrange, Mémoire sur la théorie des variations des éléments des planètes, Mémoires de la classe des sciences mathématiques et physiques de l’institut de France, 1808, pp. 1-72.
[23] J. L. Lagrange, Second mémoire sur la théorie de la variation des constantes arbitraires dans les problèmes de mécanique, Mémoires de la classe des sciences mathématiques et physiques de l’institut de France 1809, pp. 343-352.
[24] Robert G. Littlejohn, A guiding center Hamiltonian: a new approach, J. Math. Phys. 20 (1979), no. 12, 2445 – 2458. · Zbl 0444.70020 · doi:10.1063/1.524053 · doi.org
[25] G.-M. Marle, Symplectic manifolds, dynamical groups, and Hamiltonian mechanics, Differential geometry and relativity, Reidel, Dordrecht, 1976, pp. 249 – 269. Mathematical Phys. and Appl. Math., Vol. 3. · Zbl 0369.53042
[26] Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121 – 130. · Zbl 0327.58005
[27] V. P. Maslov, Théorie des perturbations et méthodes asymptotiques, Dunod, Gauthier-Villars, Paris, 1972. (Translation of 1965 Russian edition.) · Zbl 0247.47010
[28] R. B. Melrose, Equivalence of glancing hypersurfaces, Invent. Math. 37 (1976), no. 3, 165 – 191. · Zbl 0354.53033 · doi:10.1007/BF01390317 · doi.org
[29] R. B. Melrose, Forward scattering by a convex obstacle, Comm. Pure Appl. Math. 33 (1980), no. 4, 461 – 499. · Zbl 0435.35066 · doi:10.1002/cpa.3160330402 · doi.org
[30] Kenneth R. Meyer, Symmetries and integrals in mechanics, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 259 – 272.
[31] A. S. Miščenko and A. T. Fomenko, Euler equation on finite-dimensional Lie groups, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 2, 396 – 415, 471 (Russian). · Zbl 0383.58006
[32] J. Moser, A fixed point theorem in symplectic geometry, Acta Math. 141 (1978), no. 1 – 2, 17 – 34. · Zbl 0382.53035 · doi:10.1007/BF02545741 · doi.org
[33] Linda Preiss Rothschild and Joseph A. Wolf, Representations of semisimple groups associated to nilpotent orbits, Ann. Sci. École Norm. Sup. (4) 7 (1974), 155 – 173 (1975). · Zbl 0307.22012
[34] Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Springer, Berlin, 1973, pp. 265 – 529. Lecture Notes in Math., Vol. 287. · Zbl 0277.46039
[35] I. E. Segal, Quantization of nonlinear systems, J. Mathematical Phys. 1 (1960), 468 – 488. · Zbl 0099.22402 · doi:10.1063/1.1703683 · doi.org
[36] Barry Simon, The classical limit of quantum partition functions, Comm. Math. Phys. 71 (1980), no. 3, 247 – 276. · Zbl 0436.22012
[37] Jȩdrzej Śniatycki, Geometric quantization and quantum mechanics, Applied Mathematical Sciences, vol. 30, Springer-Verlag, New York-Berlin, 1980.
[38] J. Śniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J. 22 (1972/73), 267 – 275. · Zbl 0237.58002 · doi:10.1512/iumj.1972.22.22021 · doi.org
[39] J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970 (French). · Zbl 0186.58001
[40] S. Sternberg, Celestial mechanics. II, W. A. Benjamin, New York, 1969. · Zbl 0293.70013
[41] Michael E. Taylor, Grazing rays and reflection of singularities of solutions to wave equations, Comm. Pure Appl. Math. 29 (1976), no. 1, 1 – 38. · Zbl 0318.35009 · doi:10.1002/cpa.3160290102 · doi.org
[42] Nolan R. Wallach, Symplectic geometry and Fourier analysis, Math Sci Press, Brookline, Mass., 1977. With an appendix on quantum mechanics by Robert Hermann; Lie Groups: History, Frontiers and Applications, Vol. V. · Zbl 0379.53010
[43] Alan Weinstein, On Maslov’s quantization condition, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974) Springer, Berlin, 1975, pp. 341 – 372. Lecture Notes in Math., Vol. 459.
[44] Alan Weinstein, Fourier integral operators, quantization, and the spectra of Riemannian manifolds, Géométrie symplectique et physique mathématique (Colloq. Internat. CNRS, No. 237, Aix-en-Provence, 1974) Éditions Centre Nat. Recherche Sci., Paris, 1975, pp. 289 – 298 (English, with French summary). With questions by W. Klingenberg and K. Bleuler and replies by the author. · Zbl 0327.58013
[45] Alan Weinstein, Lectures on symplectic manifolds, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8 – 12, 1976; Regional Conference Series in Mathematics, No. 29. · Zbl 0406.53031
[46] A. Weinstein, The symplectic ”category, ” Proc. Conf. Differential Geometric Methods in Mathematical Physics (Clausthal-Zellerfeld, 1980) (in preparation). · Zbl 0486.58017
[47] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.