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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

References [Enhancements On Off] (What's this?)

  • 1. R. Abraham, Marsden, J., Foundations of Mechanics, Perseus Publishing, 2nd revised edition, 1994. MR 81e:58025
  • 2. B. Alberts, et al., Molecular Biology of the Cell, Garland Pub., 4th ed., 2002.
  • 3. A. Alekseev, On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom. 45 (1997), 241 - 256. MR 99b:58086
  • 4. A. Alekseev, Y. Kosmann-Schwarzbach, Manin pairs and moment maps, J. Differential Geom. 56 (2000), 133 - 165. MR 2002m:53128
  • 5. A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), 445 - 495. MR 99k:58062
  • 6. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer Verlag, 1978. MR 57:14033b
  • 7. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111:1 (1978), 61-110. MR 58:14737a; Deformation theory and quantization. II. Physical applications. Ann. Physics111:1 (1978), 111-151. MR 58:14737b
  • 8. R. Beals, P. Deift, C. Tomei, Direct and Inverse Scattering on the Line, AMS Mathematical Surveys and Monographs 28, 1988. MR 90a:58064
  • 9. M. Bertelson, Foliations Associated to Regular Poisson Structures, Commun. Contemp. Math. 3 (2001), 441 - 456. MR 2002i:53110
  • 10. A. M. Bloch, P. E. Crouch, Newton's laws and integrability of nonholonomic systems, SIAM J. Control Optim. 36 (1998), 2020 - 2039. MR 2000g:70020
  • 11. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. Anal. 136 (1996), 21 - 99. MR 97i:58048
  • 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. A. Cannas da Silva, A. Weinstein, Geometric Models for Noncommutative Algebras, AMS, Providence, RI, USA, 1999. MR 2001m:58013
  • 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. A. S. Cattaneo, G. Felder, A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), 591-611. MR 2002b:53141
  • 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. S. S. Chern, Moving frames, The mathematical heritage of Élie Cartan (Lyon, 1984), Astérisque, Numero hors Serie (1985), 67 - 77. MR 87h:58006
  • 22. J. Clemente-Gallardo, B. M. Mashke, A. van der Schaft, Kinematical constraints and algebroids, Rep. Math. Phys. 47 (2001), 415 - 429. MR 2002d:37088
  • 23. J. Conn, Normal forms for analytic Poisson structures, Annals of Math. 119 (1984), 577 - 601. MR 85f:58037
  • 24. J. Conn, Normal forms for smooth Poisson structures, Annals of Math. 121 (1985), 565 - 593. MR 86m:58050
  • 25. A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994.
  • 26. A. Coste, P. Dazord, A. Weinstein, Groupoides symplectiques, Pub. Dép. Math. Nouvelle Série. A, 2:i-ii, 1-62, Publ. Dp. Math. Nouvelle Sr. A, 87-2, Univ. Claude-Bernard, Lyon, 1987. MR 90g:58033
  • 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. R. Cushman, L. Bates, Global Aspects of Classic Integrable Systems, Birkhauser, Basel, 1997. MR 98a:58083
  • 32. R. Cushman, J. Sniatycki, eds., Proceedings of the Pacific Institute of Mathematical Sciences workshop on nonholonomic constraints in dynamics, Rep. Math. Physics 42:1/2, 1998. MR 99i:58004
  • 33. V. G. Drinfel'd, Quantum groups, Proc. ICM Berkeley, 1986, AMS, Providence, 1987, 789 - 820. MR 89f:17017
  • 34. V. G. Drinfel'd, On Poisson homogeneous spaces of Poisson Lie groups, Theor. Math. Phys. 95 (1993), 524 - 525. MR 94k:58045
  • 35. J.-P. Dufour, A. Wade, Formes normales de structures de Poisson ayant un 1-jet nul en un point, J. Geom. Phys. 26 (1998), 79 - 96. MR 99d:58152
  • 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. S. Evens, J.-H. Lu, Poisson harmonic forms, Kostant harmonic forms, and $S^1$-equivariant cohomology of $K/T$, Adv. Math. 142 (1999), 171 - 220. MR 2001e:53085
  • 38. S. Evens, J.-H. Lu, A. Weinstein, Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quart. J. Math. 50 (1999), 417 - 436. MR 2000i:53114
  • 39. R. L. Fernandes, Connections in Poisson geometry: holonomy and invariants, J. Differential Geom. 54 (2000), 303 - 356. MR 2001m:53152
  • 40. R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. in Math. 170 (2002), 119-179.
  • 41. M. Frémond, Rigid bodies collisions, Phis. Lett. A 204:7 (1995), 33 - 41. MR 96i:70012
  • 42. M. Frémond, Internal constraints in mechanics, Philos. T. Roy. Soc. A 359:1789 (2001), 2309 - 2326. MR 2002k:74003
  • 43. R. Gardner, The Method of Equivalence and its Applications, SIAM, 1989. MR 91j:58007
  • 44. M. Gerstenhaber, On the deformations of rings and algebras, Ann. of Math. 2:79 (1964) 59-103. MR 30:2034
  • 45. V. L. Ginzburg, Momentum mappings and Poisson cohomology, Internat. J. Math. 7 (1996), 329 - 358. MR 97g:17022
  • 46. V. L. Ginzburg, Grothendieck groups of Poisson vector bundles, to appear in J. Symplectic Geom.
  • 47. V. L. Ginzburg, J.-H. Lu, Poisson cohomology of Morita-equivalent Poisson manifolds, Duke Math. J. 68 (1992), 199 - 205. MR 93k:58091
  • 48. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer Applied Mathematical Sciences, Vol 42), Springer-Verlag, 1990. MR 93e:58046
  • 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. J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57 - 113. MR 92e:17027
  • 52. J. Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103 - 159. MR 2000f:53109
  • 53. A. Ibort, M. de Leon, J. C. Marrero, D. M. Diego, Dirac brackets in constrained dynamics, Fortschr. Phys. 47 (1999), 459 - 492. MR 2000d:37076
  • 54. C. Kane, J. E. Marsden, M. Ortiz, M. West, Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems, Int. J. Num. Math. Eng. 49 (2000), 1295 - 1325. MR 2001k:37132
  • 55. J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal. 118 (1992), 113 - 148. MR 93e:70005
  • 56. J. Koiller, R. Montgomery, K. Ehlers, Problems and progress in Microswimming, J. Nonlinear Sci. 6 (1996), 507 - 541. MR 98b:76083
  • 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, Sov. Phys. Dokl. 28:9 (1983), 735 - 737. MR 84m:70024
  • 61. N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics. Springer-Verlag, New York, 1998. MR 2000g:81081
  • 62. A. D. Lewis, The geometry of the Gibbs-Appell equations and Gauss's Principle of Least Constraint, Rep. Math. Phys. (1996), 38:1, 11 - 28. MR 97g:58048
  • 63. A. D. Lewis, R. M. Murray, Configuration controllability of simple mechanical control systems, SIAM Journal on Control and Optimization 35:3 (1997), 766 - 790. MR 99a:70045
  • 64. A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geom. 12 (1977), 253 - 300. MR 58:18565
  • 65. Z.-J. Liu, A. Weinstein, P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547 - 574. MR 98f:58203
  • 66. J.-H. Lu, A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions , J. Differential Geom. 31 (1990), 501 - 526. MR 91c:22012
  • 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, J. D. Meiss (Compilers), Hamiltonian Dynamical Systems: A Reprint Selection, Adam Hilger, 1987. MR 92g:58028
  • 69. C. Marle, P. Libermann, Symplectic Geometry and Analytical Mechanics (Mathematics and Its Applications, 35), D Reidel Pub. Co, 1987. MR 88c:58016
  • 70. J. E. Marsden, Mechanics, Dynamics and Symmetry, IAS/Park City Mathematics Series 7, 1999. MR 2001j:70001
  • 71. J. E. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (Texts in Applied Mathematics, 17), Springer Verlag; 2nd edition, 1999. MR 2000i:70002
  • 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. R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society Mathematical Surveys and Monographs 91, 2002. MR 2002m:53045
  • 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. J.-P. Ortega, T. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity 12 (1999), 693 - 720. MR 2000g:37075
  • 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. L. Perko, Differential Equations and Dynamical Systems (Springer Texts in Applied Mathematics, 7), Springer Verlag, 3rd edition, 2001. MR 2001k:34001
  • 81. H. Poincaré, Sur une forme nouvelle des équations de la Mécanique, C. R. Acad. Sci. Paris 132 (1901), 369 - 371.
  • 82. A. Ruina, Nonholonomic Stability Aspects of Piecewise-Holonomic Systems Rep. Math. Phys. 42:1/2 (1998), 91 - 100. MR 99k:70017
  • 83. P. Schaller, T. Strobl, Poisson structure induced (topological) field theories, Modern Phys. Lett. A 9:33 (1994), 3129-3136. MR 96d:81210
  • 84. S. Smale, M. Hirsch, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1997. MR 58:6484
  • 85. A. Sommerfeld, Lectures on Theoretical Physics: Mechanics, Academic Press, 1952.
  • 86. M. Spivak, Comprehensive Introduction to Differential Geometry (Volumes 1-5), Publish or Perish, 1979. MR 82g:53003a, MR 82g:53003b, MR 82g:53003c, MR 82g:53003d, MR 82g:53003e
  • 87. S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books, 2001.
  • 88. I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, Basel, 1994. MR 95h:58057
  • 89. A. van der Schaft, B. M. Mashke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys. 34 (1994), 225 - 233. MR 95k:70031
  • 90. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, (Springer Universitexts), Springer Verlag, 2nd Rev edition, 1997. MR 97g:34003
  • 91. A. M. Vershik, V. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, in V. I. Arnold, S. P. Novikov (Eds.), Dynamical Systems VII, Springer-Verlag, 1994. MR 88g:58004, MR 89f:58007
  • 92. A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), 523 - 557. MR 87k:58099
  • 93. A. Weinstein, Symplectic groupoids and irrational rotation algebras, in Symplectic geometry, groupoids, and integrable systems: Papers from the Sixth South-Rhône Geometry Seminar held in Berkeley, California, May 22-June 2, 1989, ed. by P. Dazord and A. Weinstein, MSRI Publications, 20, Springer-Verlag, New York, 1991. MR 92h:46103
  • 94. A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379 - 394. MR 98k:58095
  • 95. A. Weinstein, Poisson geometry, Differential Geom. Appl. 9 (1998), 213 - 238. MR 99i:58066
  • 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. P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545 - 560. MR 2000b:17025
  • 99. D. M. Zenkov, A. M. Bloch, J. E. Marsden, The energy-momentum method for the stability of nonholonomic systems, Dyn. Stab. Systems 13:2 (1998), 123 - 165. MR 2000f:70016
  • 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
MSC (2000): Primary 70-00, 70-01, 70E55, 70F20, 70G75
Published electronically: April 9, 2003
Additional Notes: This research was supported in part by a CNPq fellowship
Review copyright: © Copyright 2003 American Mathematical Society
American Mathematical Society