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Book Review

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Book Information:

Author: John G. Papastavridis
Title: Analytical mechanics: A comprehensive treatise on the dynamics of constrained systems; for engineers, physicists and mathematicians
Additional book information: Oxford University Press, 2002, xxii + 1392 pp., ISBN 0-19-512697-1, $295.00

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  • 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. MR 515141
  • 2. B. Alberts, et al., Molecular Biology of the Cell, Garland Pub., 4th ed., 2002.
  • 3. A. Yu. Alekseev, On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom. 45 (1997), no. 2, 241–256. MR 1449971
  • 4. Anton Alekseev and Yvette Kosmann-Schwarzbach, Manin pairs and moment maps, J. Differential Geom. 56 (2000), no. 1, 133–165. MR 1863024
  • 5. Anton Alekseev, Anton Malkin, and Eckhard Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), no. 3, 445–495. MR 1638045
  • 6. V. I. Arnold, 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. MR 0690288
  • 7. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1, 61–110. MR 0496157
    F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), no. 1, 111–151. MR 0496158
  • 8. Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, Mathematical Surveys and Monographs, vol. 28, American Mathematical Society, Providence, RI, 1988. MR 954382
  • 9. Mélanie Bertelson, Foliations associated to regular Poisson structures, Commun. Contemp. Math. 3 (2001), no. 3, 441–456. MR 1849650, 10.1142/S0219199701000445
  • 10. A. M. Bloch and P. E. Crouch, Newton’s law and integrability of nonholonomic systems, SIAM J. Control Optim. 36 (1998), no. 6, 2020–2039 (electronic). MR 1638936, 10.1137/S0363012995291634
  • 11. Anthony M. Bloch, P. S. Krishnaprasad, Jerrold E. Marsden, and Richard M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal. 136 (1996), no. 1, 21–99. MR 1423003, 10.1007/BF02199365
  • 12. D. Boal, Mechanics of the Cell, Cambridge University Press, 2002.
  • 13. H. Bursztyn, O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, to appear in Ann. Inst. Fourier.
  • 14. H. Bursztyn, S. Waldmann, The characteristic classes of Morita equivalent star products on symplectic manifolds, Comm. Math. Phys. 228 (2002), 103-121.
  • 15. H. Cabral, F. Diacu, Classical and Celestial Mechanics: The Recife Lectures, Princeton University Press, 2002.
  • 16. Ana Cannas da Silva and Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, vol. 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999. MR 1747916
  • 17. É. Cartan, Sur la represéntation géométrique des systèmes matériels nonholonomes, Proc. Int. Congr. Math. 4, Bologna (1928), 253 - 261.
  • 18. É. Cartan, Selecta, Gauthier-Villars, 1910; Ann. Sci. École Norm. Sup. 27 (1939), 109 - 192.
  • 19. Alberto S. Cattaneo and Giovanni Felder, A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), no. 3, 591–611. MR 1779159, 10.1007/s002200000229
  • 20. A. S. Cattaneo, G. Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, ed. N. P. Landsman et al., Progress in Mathematics 198 (2001), 41-73.
  • 21. Shiing Shen Chern, Moving frames, Astérisque Numero Hors Serie (1985), 67–77. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837194
  • 22. J. Clemente-Gallardo, B. Maschke, and A. J. van der Schaft, Kinematical constraints and algebroids, Rep. Math. Phys. 47 (2001), no. 3, 413–429. MR 1847637, 10.1016/S0034-4877(01)80053-7
  • 23. Jack F. Conn, Normal forms for analytic Poisson structures, Ann. of Math. (2) 119 (1984), no. 3, 577–601. MR 744864, 10.2307/2007086
  • 24. Jack F. Conn, Normal forms for smooth Poisson structures, Ann. of Math. (2) 121 (1985), no. 3, 565–593. MR 794374, 10.2307/1971210
  • 25. A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994.
  • 26. A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, vol. 87, Univ. Claude-Bernard, Lyon, 1987, pp. i–ii, 1–62 (French). MR 996653
  • 27. M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, to appear in Comment. Math. Helv.
  • 28. M. Crainic, R. Fernandes, Integrability of Poisson brackets, Preprint math.DG/0210152.
  • 29. M. Crainic, R. L. Fernandes, Integrability of Lie brackets, to appear in Annals of Math.
  • 30. M. Csete, J. Doyle, Reverse engineering of biological complexity, Science 295 (2002), 1664 - 1669; see also the site http://www.cds.caltech.edu/doyle/CmplxNets/
  • 31. Richard H. Cushman and Larry M. Bates, Global aspects of classical integrable systems, Birkhäuser Verlag, Basel, 1997. MR 1438060
  • 32. Richard Cushman and Jędrzej Śniatycki (eds.), Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics, Pergamon Press, Oxford, 1998. Rep. Math. Phys. 42 (1998), no. 1-2. MR 1675475
  • 33. V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
  • 34. V. G. Drinfel′d, On Poisson homogeneous spaces of Poisson-Lie groups, Teoret. Mat. Fiz. 95 (1993), no. 2, 226–227 (English, with English and Russian summaries); English transl., Theoret. and Math. Phys. 95 (1993), no. 2, 524–525. MR 1243249, 10.1007/BF01017137
  • 35. J.-P. Dufour and A. Wade, Formes normales de structures de Poisson ayant un 1-jet nul en un point, J. Geom. Phys. 26 (1998), no. 1-2, 79–96 (French, with English and French summaries). MR 1626032, 10.1016/S0393-0440(97)00039-9
  • 36. K. Ehlers, Geometric equivalence on nonholonomic three dimensional contact manifolds, to appear in Proceed. Fourth Intl. Conf. on Dynamical Systems and Differential Equations, Wilmington, NC, May 24-27, 2002.
  • 37. Sam Evens and Jiang-Hua Lu, Poisson harmonic forms, Kostant harmonic forms, and the 𝑆¹-equivariant cohomology of 𝐾/𝑇, Adv. Math. 142 (1999), no. 2, 171–220. MR 1680047, 10.1006/aima.1998.1788
  • 38. Sam Evens, Jiang-Hua Lu, and Alan Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 417–436. MR 1726784, 10.1093/qjmath/50.200.417
  • 39. Rui Loja Fernandes, Connections in Poisson geometry. I. Holonomy and invariants, J. Differential Geom. 54 (2000), no. 2, 303–365. MR 1818181
  • 40. R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. in Math. 170 (2002), 119-179.
  • 41. Michel Frémond, Rigid bodies collisions, Phys. Lett. A 204 (1995), no. 1, 33–41. MR 1346712, 10.1016/0375-9601(95)00418-3
  • 42. Michel Frémond, Internal constraints in mechanics, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), no. 1789, 2309–2326. Non-smooth mechanics. MR 1884302, 10.1098/rsta.2001.0853
  • 43. Robert B. Gardner, The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1062197
  • 44. Murray Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59–103. MR 0171807
  • 45. Viktor L. Ginzburg, Momentum mappings and Poisson cohomology, Internat. J. Math. 7 (1996), no. 3, 329–358. MR 1395934, 10.1142/S0129167X96000207
  • 46. V. L. Ginzburg, Grothendieck groups of Poisson vector bundles, to appear in J. Symplectic Geom.
  • 47. Viktor L. Ginzburg and Jiang-Hua Lu, Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices 10 (1992), 199–205. MR 1191570, 10.1155/S1073792892000229
  • 48. John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original. MR 1139515
  • 49. J. Howard, Mechanics of motor proteins and the cytoskeleton, Sinauer Associates, Publishers, Sunderland, Mass., 2001.
  • 50. G. Huber, R. E. Goldstein, A. Goriely, C. W. Wolgemuth, Bistable helices, Phys. Rev. Lett. 84 (2000) 1631 -1634.
  • 51. Johannes Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113. MR 1058984, 10.1515/crll.1990.408.57
  • 52. Johannes Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103–159. MR 1696093, 10.1515/crll.1999.043
  • 53. Alberto Ibort, Manuel de Leon, Juan C. Marrero, and David Martin de Diego, Dirac brackets in constrained dynamics, Fortschr. Phys. 47 (1999), no. 5, 459–492. MR 1692290, 10.1002/(SICI)1521-3978(199906)47:5<459::AID-PROP459>3.0.CO;2-E
  • 54. C. Kane, J. E. Marsden, M. Ortiz, and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Internat. J. Numer. Methods Engrg. 49 (2000), no. 10, 1295–1325. MR 1805500, 10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.3.CO;2-N
  • 55. Jair Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal. 118 (1992), no. 2, 113–148. MR 1158932, 10.1007/BF00375092
  • 56. J. Koiller, K. Ehlers, and R. Montgomery, Problems and progress in microswimming, J. Nonlinear Sci. 6 (1996), no. 6, 507–541. MR 1419447, 10.1007/s003329900021
  • 57. J. Koiller, P. M. Rios, Nonholonomic systems with symmetry allowing a conformally symplectic reduction, Proc. IV Int. Symp. of Hamiltonian Systems and Celestial Mechanics (Mexico 2001), edited by J. Delgado, E. A. Lacomba, E. Perez-Chavela, to appear.
  • 58. J. Koiller, P. M. Rios, K. Ehlers, Moving frames for cotangent bundles, Rep. Math. Phys. 49:2/3 (2002), 225 - 238.
  • 59. M. Kontsevich, Deformation quantization of Poisson manifolds, Preprint 1997, q-alg/9709040.
  • 60. V. V. Kozlov, Realization of nonintegrable constraints in classical mechanics, Dokl. Akad. Nauk SSSR 272 (1983), no. 3, 550–554 (Russian). MR 723778
  • 61. N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998. MR 1662141
  • 62. Andrew D. Lewis, The geometry of the Gibbs-Appell equations and Gauss’ principle of least constraint, Rep. Math. Phys. 38 (1996), no. 1, 11–28. MR 1414422, 10.1016/0034-4877(96)87675-0
  • 63. Andrew D. Lewis and Richard M. Murray, Configuration controllability of simple mechanical control systems, SIAM J. Control Optim. 35 (1997), no. 3, 766–790. MR 1444338, 10.1137/S0363012995287155
  • 64. André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), no. 2, 253–300 (French). MR 0501133
  • 65. Zhang-Ju Liu, Alan Weinstein, and Ping Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), no. 3, 547–574. MR 1472888
  • 66. Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. MR 1037412
  • 67. H. Mabuchi, The Quantum-Classical Transition on Trial: Is the Whole More than the Sum of the Parts? (popular article), Caltech Engin. and Sci. Magazine 65:2 (2002).
  • 68. R. S. MacKay and J. D. Meiss (eds.), Hamiltonian dynamical systems, Adam Hilger, Ltd., Bristol, 1987. MR 1103556
  • 69. Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications, vol. 35, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the French by Bertram Eugene Schwarzbach. MR 882548
  • 70. Jerrold E. Marsden, Park City lectures on mechanics, dynamics, and symmetry, Symplectic geometry and topology (Park City, UT, 1997) IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, 1999, pp. 335–430. MR 1702948
  • 71. Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, 2nd ed., Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1999. A basic exposition of classical mechanical systems. MR 1723696
  • 72. J. E. Marsden, M. West, Discrete mechanics and variational integrators, Acta Numerica (2001), 357 - 514.
  • 73. J. C. Monforte, Geometric, Control, and Numerical Aspects of Nonholonomic Systems, Springer-Verlag Lecture Notes in Mathematics 1793, 2002.
  • 74. Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362
  • 75. R. Murray, NSF Panel on Future Directions in Control, Dynamics, and Systems, 2002 (http://www.cds.caltech.edu/murray/cdspanel/).
  • 76. R. Murray, M. Teixeira, Control and Dynamical Systems Alliance, Fipse/Capes proposal (http://www.ed.gov/offices/OPE/FIPSE/Brazil/Brazil2002awards.html).
  • 77. J. I. Neimark, N. A. Fufaev, Dynamics of nonholonomic systems, AMS, Providence, 1972.
  • 78. Juan-Pablo Ortega and Tudor S. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity 12 (1999), no. 3, 693–720. MR 1690200, 10.1088/0951-7715/12/3/315
  • 79. J. Ostrowski, J. Burdick, A. D. Lewis, R. M. Murray, The mechanics of undulatory locomotion: the mixed kinematic and dynamic case, IEE Conf. on Robotics and Automation (1995), 1945 - 1951.
  • 80. Lawrence Perko, Differential equations and dynamical systems, 3rd ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001. MR 1801796
  • 81. H. Poincaré, Sur une forme nouvelle des équations de la Mécanique, C. R. Acad. Sci. Paris 132 (1901), 369 - 371.
  • 82. Andy Ruina, Nonholonomic stability aspects of piecewise holonomic systems, Rep. Math. Phys. 42 (1998), no. 1-2, 91–100. Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics (Calgary, AB, 1997). MR 1656277, 10.1016/S0034-4877(98)80006-2
  • 83. Peter Schaller and Thomas Strobl, Poisson structure induced (topological) field theories, Modern Phys. Lett. A 9 (1994), no. 33, 3129–3136. MR 1303989, 10.1142/S0217732394002951
  • 84. Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 60. MR 0486784
  • 85. A. Sommerfeld, Lectures on Theoretical Physics: Mechanics, Academic Press, 1952.
  • 86. Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532830
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532831
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532832
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532833
    Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532834
  • 87. S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books, 2001.
  • 88. Izu Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. MR 1269545
  • 89. A. J. van der Schaft and B. M. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys. 34 (1994), no. 2, 225–233. MR 1323130, 10.1016/0034-4877(94)90038-8
  • 90. Ferdinand Verhulst, Nonlinear differential equations and dynamical systems, 2nd ed., Universitext, Springer-Verlag, Berlin, 1996. Translated from the 1985 Dutch original. MR 1422255
  • 91. R. V. Gamkrelidze (ed.), Sovremennye problemy matematiki. Fundamentalnye napravleniya. Tom 16, \cyr Itogi Nauki i Tekhniki. [Progress in Science and Technology], Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987 (Russian). Dinamicheskie sistemy. 7. [Dynamical systems. 7]. MR 922069
    A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems. Geometry of distributions and variational problems, Current problems in mathematics. Fundamental directions, Vol. 16 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, pp. 5–85, 307 (Russian). MR 922070
  • 92. Alan Weinstein, Errata and addenda: “The local structure of Poisson manifolds” [J. Differential Geom. 18 (1983), no. 3, 523–557; MR0723816 (86i:58059)], J. Differential Geom. 22 (1985), no. 2, 255. MR 834280
  • 93. Alan Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1991, pp. 281–290. MR 1104934, 10.1007/978-1-4613-9719-9_19
  • 94. Alan Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), no. 3-4, 379–394. MR 1484598, 10.1016/S0393-0440(97)80011-3
  • 95. Alan Weinstein, Poisson geometry, Differential Geom. Appl. 9 (1998), no. 1-2, 213–238. Symplectic geometry. MR 1636305, 10.1016/S0926-2245(98)00022-9
  • 96. A. Weinstein, From Riemann geometry to Poisson geometry and back again (lecture at Chern Symposium, MSRI, March 6, 1998) (www.msri.org/publications/ln/ msri/1998/chern/weinstein/1/index.html).
  • 97. A. Weinstein, J. E. Marsden, Comments on the history, theory, and applications of symplectic reduction, in Schlichenmaier, M., et al., eds., Quantization of Singular Symplectic Quotients, Birkhauser, 2001.
  • 98. Ping Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), no. 3, 545–560. MR 1675117, 10.1007/s002200050540
  • 99. Dmitry V. Zenkov, Anthony M. Bloch, and Jerrold E. Marsden, The energy-momentum method for the stability of non-holonomic systems, Dynam. Stability Systems 13 (1998), no. 2, 123–165. MR 1629279, 10.1080/02681119808806257
  • 100. N. Zung, Levi decomposition of analytic Poisson structures and Lie algebroids, to appear in Topology.

Review Information:

Reviewer: Jair Koiller
Email: jkoiller@fgv.br
Journal: Bull. Amer. Math. Soc. 40 (2003), 405-419
Published electronically: April 9, 2003
Additional Notes: This research was supported in part by a CNPq fellowship
Review copyright: © Copyright 2003 American Mathematical Society