Book Review
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Book Information:
Authors:
J.-P. Ortega and
T. S. Ratiu
Title:
Momentum maps and Hamiltonian reduction
Additional book information:
Progress in Math., vol. 222, Birkhäuser Boston, Inc.,
Boston, MA,
2004,
xxxiv + 497 pp.,
ISBN 0-8176-4307-9,
$69.95$
1. N. Aronszajn, ``Subcartesian and subriemannian spaces'', Notices A.M.S. 14 (1967) 111.
Judith Arms, Reduction of Poisson algebras at nonzero momentum values, J. Geom. Phys. 21 (1996), no. 1, 81–95. MR 1421256, DOI 10.1016/S0393-0440(96)00009-5
Judith M. Arms, Mark J. Gotay, and George Jennings, Geometric and algebraic reduction for singular momentum maps, Adv. Math. 79 (1990), no. 1, 43–103. MR 1031826, DOI 10.1016/0001-8708(90)90058-U
Peter G. Bergmann and Irwin Goldberg, Dirac bracket transformations in phase space, Phys. Rev. (2) 98 (1955), 531–538. MR 74314
Edward Bierstone, Lifting isotopies from orbit spaces, Topology 14 (1975), no. 3, 245–252. MR 375356, DOI 10.1016/0040-9383(75)90005-1
Richard Cushman, Reduction, Brouwer’s Hamiltonian, and the critical inclination, Celestial Mech. 31 (1983), no. 4, 401–429. MR 737806, DOI 10.1007/BF01230294
Richard Cushman and Jȩdrzej Śniatycki, Differential structure of orbit spaces, Canad. J. Math. 53 (2001), no. 4, 715–755. MR 1848504, DOI 10.4153/CJM-2001-029-1
P. A. M. Dirac, Generalized Hamiltonian dynamics, Canad. J. Math. 2 (1950), 129–148. MR 43724, DOI 10.4153/cjm-1950-012-1
Avner Friedman, Generalized functions and partial differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0165388
V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
11. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems, Accad. Nazionale dei Lincei, Rome, 1976.
H. P. Künzle, Degenerate Lagrangian systems, Ann. Inst. H. Poincaré Sect. A (N.S.) 11 (1969), 393–414 (English, with French summary). MR 278586
André Lichnerowicz, Variété symplectique et dynamique associée à une sous-variété, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), A523–A527. MR 405503
Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR 402819, DOI 10.1016/0034-4877(74)90021-4
Eckhard Meinrenken and Reyer Sjamaar, Singular reduction and quantization, Topology 38 (1999), no. 4, 699–762. MR 1679797, DOI 10.1016/S0040-9383(98)00012-3
16. J.-P. Ortega, Symmetry, Reduction and Stability in Hamiltonian Systems, Ph.D. Thesis, University of California, Santa Cruz, 1998.
Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI 10.2307/1970335
Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68. MR 370643, DOI 10.1016/0040-9383(75)90036-1
19. R. Sikorski, Wstep do Geometrii Rózniczkowej, PWN, Warszawa, 1972.
Jędrzej Śniatycki, Dirac brackets in geometric dynamics, Ann. Inst. H. Poincaré Sect. A (N.S.) 20 (1974), 365–372. MR 358860
Jȩdrzej Śniatycki, Orbits of families of vector fields on subcartesian spaces, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7, 2257–2296 (English, with English and French summaries). MR 2044173
Jędrzej Śniatycki and Alan Weinstein, Reduction and quantization for singular momentum mappings, Lett. Math. Phys. 7 (1983), no. 2, 155–161. MR 708438, DOI 10.1007/BF00419934
P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3) 29 (1974), 699–713. MR 362395, DOI 10.1112/plms/s3-29.4.699
24. W.M. Tulczyjew, unpublished lectures, Institute of Theoretical Physics, Warsaw University, 1967.
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
- 1.
- N. Aronszajn, ``Subcartesian and subriemannian spaces'', Notices A.M.S. 14 (1967) 111.
- 2.
- J.M. Arms, ``Reduction of Poisson algebras at nonzero momentum values'', J. Geom. Phys. 21 (1996) 81-95. MR 1421256
- 3.
- J.M. Arms, M. Gotay and G. Jennings, ``Geometric and algebraic reduction for singular momentum maps'', Adv. Math. 79 (1990) 43-103. MR 1031826
- 4.
- P.G. Bergmann and I. Goldberg, ``Dirac bracket transformations in phase space'', Phys. Rev. 98 (1955) 531-538. MR 0074314
- 5.
- E. Bierstone, ``Lifting isotopies from orbit spaces'', Topology 14 (1975) 245-252. MR 0375356
- 6.
- R. Cushman, ``Reduction, Brouwer's Hamiltonian and the critical inclination'', Celest. Mech. 31 (1983) 401-429. MR 0737806
- 7.
- R. Cushman and J. Sniatycki, ``Differential structure of orbit spaces'', Canad. J. Math. 53 (2001) 715-755. MR 1848504
- 8.
- P.A.M. Dirac, ``Generalized Hamiltonian dynamics'', Canad. J. Math. 2 (1950) 129-148. MR 0043724
- 9.
- A. Friedman, Generalized Functions and Partial Differential Equations, Prentice Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0165388
- 10.
- V.W. Guillemin and S. Sternberg, ``Geometric quantization and multiplicities of group representations'', Invent. Math. 67 (1982) 515-538. MR 0664118
- 11.
- A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems, Accad. Nazionale dei Lincei, Rome, 1976.
- 12.
- H. Künzle, ``Degenerate Lagrangian systems'', Ann. Inst. H. Poincaré 11 (1969) 393-414. MR 0278586
- 13.
- A. Lichnerowicz, ``Variété symplectique et dynamique associée à une sous-variété'', C.R. Acad. Sci. Paris, Sér. A 280 (1975) 523-527. MR 0405503
- 14.
- J.E. Marsden and A. Weinstein, ``Reduction of symplectic manifolds with symmetry'', Rep. Math. Phys. 5 (1974) 121-130. MR 0402819
- 15.
- E. Meinrenken and R. Sjamaar, ``Singular reduction and quantization'', Topology 38 (1999) 699-762. MR 1679797
- 16.
- J.-P. Ortega, Symmetry, Reduction and Stability in Hamiltonian Systems, Ph.D. Thesis, University of California, Santa Cruz, 1998.
- 17.
- R. Palais, ``On the existence of slices for actions of non-compact Lie groups'', Ann. Math. 73 (1961) 295-323. MR 0126506
- 18.
- G.W. Schwarz, ``Smooth functions invariant under the action of a compact Lie group'', Topology 14 (1975) 63-68. MR 0370643
- 19.
- R. Sikorski, Wstep do Geometrii Rózniczkowej, PWN, Warszawa, 1972.
- 20.
- J. Sniatycki, ``Dirac brackets in geometric dynamics'', Ann. Inst. Henri Poincaré 20 (1974) 365-372. MR 0358860
- 21.
- J. Sniatycki, ``Orbits of families of vector fields on subcartesian spaces'', Ann. Inst. Fourier (Grenoble) 53 (2003) 2257-2296. MR 2044173
- 22.
- J. Sniatycki and A. Weinstein, ``Reduction and quantization for singular momentum mappings'', Lett. Math. Phys. 7 (1983) 155-161. MR 0708438
- 23.
- P. Stefan, ``Accessible sets, orbits and foliations with singularities'', Proc. London Math. Soc. 29 (1974) 699-713. MR 0362395
- 24.
- W.M. Tulczyjew, unpublished lectures, Institute of Theoretical Physics, Warsaw University, 1967.
- 25.
- H. Weyl, The Classical Groups (2nd ed.), Princeton University Press, Princeton, 1946. MR 1488158
Review Information:
Reviewer:
Jȩdrzej Śniatycki
Affiliation:
University of Calgary
Email:
sniat@math.ucalgary.ca
Journal:
Bull. Amer. Math. Soc.
42 (2005), 215-223
Published electronically:
November 29, 2004
Review copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.