Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Author: Robert Hermann
Title: Cartanian geometry, nonlinear waves, and control theory
Additional book information: Interdisciplinary Mathematics, volumes 20 and 21, Math. Sci. Press, Brookline, Mass., Part A, 1979, xv + 501 pp., $50.00; Part B, 1980, xii + 585 pp., $60.00.

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

  • 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. V. I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Math., No. 60, Springer-Verlag, New York, 1978. MR 690288
  • 3. K. Bleuler and A. Reetz (Editors), Differential geometric methods in mathematical physics. I, Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin-Heidelberg, 1977. MR 434688
  • 4. Konrad Bleuler and Axel Reetz (eds.), Differential geometrical methods in mathematical physics. II, Lecture Notes in Mathematics, vol. 676, Springer, Berlin, 1978. MR 519605
  • 5. Frederick Bloom, Modern differential geometric techniques in the theory of continuous distributions of dislocations, Lecture Notes in Mathematics, vol. 733, Springer, Berlin, 1979. MR 546839
  • 6. R. W. Brockett, Finite dimensional linear systems, Wiley, New York, 1969.
  • 7. R. W. Brockett, Systems theory on group manifolds and coset spaces, SIAM J. Control Optim. 10 (1972), 265-284. MR 315559
  • 8. R. W. Brockett, Lie theory and control systems defined on spheres, SIAM J. Appl. Math. 25 (1973), 213-225. MR 327337
  • 9. R. W. Brockett, Nonlinear systems and differential geometry, Proc. of the IEEE 64 (1976), 61-72. MR 432255
  • 10. P. Brunovsky, A classification of linear controllable systems, Kybernetika (Prague) 6 (3), 1970, 173-188. MR 284247
  • 11. Christopher I. Byrnes (ed.), Partial differential equations and geometry, Lecture Notes in Pure and Applied Mathematics, vol. 48, Marcel Dekker, Inc., New York, 1979. MR 535586
  • 12. Christopher I. Byrnes and Clyde F. Martin (eds.), Geometrical methods for the theory of linear systems, NATO Advanced Study Institute Series. Ser. C: Mathematical and Physical Sciences, vol. 62, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1980. MR 608990
  • 13. P. Chernoff and J. Marsden, Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Math., vol. 425, Springer-Verlag, New York, 1974. MR 650113
  • 14. P. Chernoff and J. Marsden, Review of "Lie algebras and quantum mechanics" and "Vector bundles in mathematical physics" by Robert Hermann, Bull. Amer. Math. Soc. 79(1973), 1150-1162.
  • 15. P. E. Crouch, Dynamical realizations of finite Volterra series, SIAM J. Control Optim. 19 (1981), no. 2, 177–202. MR 605615, https://doi.org/10.1137/0319014
  • 16. M. Daniel and C.-M. Viallet, The geometrical setting of gauge theories of the Yang-Mills type, Rev. Modern Phys. 52 (1980), no. 1, 175–197. MR 553409, https://doi.org/10.1103/RevModPhys.52.175
  • 17. E. Dierker, Topological methods in Walrasian economics, Lecture Notes in Economics and Systems, No. 92, Springer-Verlag, Berlin, 1970. MR 378736
  • 18. J. Dieudonné, Treatise on analysis, vol. 10-IV, Academic Press, New York, 1974. MR 362066
  • 19. W. Drechsler and M. E. Mayer, Fiber bundle techniques in gauge theories, Lecture Notes in Physics, No. 67, Springer-Verlag, Berlin-Heidelberg, 1977. MR 443737
  • 20. E. J. Flaherty, Hermitian and Kählerian geometry in relativity, Lecture Notes in Phys., No. 46, Springer-Verlag, Berlin-Heidelberg, 1976. MR 411532
  • 21. Edward J. Flaherty Jr., Complex variables in relativity, General relativity and gravitation, Vol. 2, Plenum, New York-London, 1980, pp. 207–239. MR 617921
  • 22. Theodore Frankel, Gravitational curvature, W. H. Freeman and Co., San Francisco, Calif., 1979. An introduction to Einstein’s theory. MR 518868
  • 23. F. R. Gantmacher, The theory of matrices, Vol. 2, Chelsea, New York, 1959.
  • 24. R. Gunning, Lectures on vector bundles over Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1967. MR 230326
  • 25. V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys, no. 14, Amer. Math. Soc. Providence, R. I., 1977. MR 516965
  • 26. S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Univ. Press, London and New York, 1973. MR 424186
  • 27. G. W. Haynes and H. Hermes, Nonlinear controllability via Lie theory, SIAM J. Control Optim. 8 (1970), 450-460. MR 277859
  • 28. R. Hermann, Algebraic topics in systems theory, Interdisciplinary Mathematics, Vol. 3, Math-Sci Press, Brookline, Mass., 1973. MR 504083
  • 29. R. Hermann, On the accessibility problem in control theory, International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 325-332. MR 149402
  • 30. R. Hermann, Vector bundles in mathematical physics, vol. 1, Benjamin, New York, 1970. MR 266551
  • 31. R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Autom. Control. AC-22 (1977), 728-740. MR 476017
  • 32. R. Hermann and C. F. Martin, Applications of algebraic geometry to systems theory--Part I, IEEE Trans. Automat. Control AC-22 (1977), 19-25. MR 444172
  • 33. R. Hermann, Review of "Elements of soliton theory" by G. L. Lamb, Jr., Bull. Amer. Math. Soc. (N. S.) 5 (1981), 203-209.
  • 34. H. Hermes, On local and global controllability, SIAM J. Control Optim. 12 (1974), 252-261. MR 451293
  • 35. R. M. Hirschorn, Controllability in nonlinear systems, J. Differential Equations 19 (1975), 46-61. MR 431271
  • 36. R. M. Hirschorn, Invertibility of control systems on Lie groups, SIAM J. Control Optim. 15 (1977), 1034-1049. MR 527880
  • 37. R. M. Hirschorn, (𝐴,\cal𝐵)-invariant distributions and disturbance decoupling of nonlinear systems, SIAM J. Control Optim. 19 (1981), no. 1, 1–19. MR 603076, https://doi.org/10.1137/0319001
  • 38. V. Jurdjevic and H. J. Sussmann, Control systems on Lie groups, J. Differential Equations 12 (1972), 313-329. MR 331185
  • 39. V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: accessibility, J. Differential Equations 39 (1981), no. 2, 186–211. MR 607781, https://doi.org/10.1016/0022-0396(81)90072-3
  • 40. R. E. Kalman, Kronecker invariants and feedback, Conf. Ordinary Differential Equations (NRL Mathematics Research Center, June 14-23, 1971), Ordinary Differential Equations (C. Weiss, ed.), Academic Press, New York, 1972, pp. 459-471. MR 421751
  • 41. B. Kostant, Quantization and unitary representations, Lecture Notes in Math., vol. 170, Springer-Verlag, New York, 1970. MR 294568
  • 42. A. J. Krener, A decomposition theory for differentiable systems, SIAM J. Control Optim. 15 (1977), 813-829. MR 479481
  • 43. Alberto Isidori, Arthur J. Krener, Claudio Gori-Giorgi, and Salvatore Monaco, Nonlinear decoupling via feedback: a differential geometric approach, IEEE Trans. Automat. Control 26 (1981), no. 2, 331–345. MR 613540, https://doi.org/10.1109/TAC.1981.1102604
  • 44. D. E. Lerner and P. D. Sommers (eds.), Complex manifold techniques in theoretical physics, Research Notes in Mathematics, vol. 32, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 564439
  • 45. C. Lobry, Controlabilite des systemes non lineaires, SIAM J. Control Optim. 8 (1970), 573-605. MR 271979
  • 46. S. Mac Lane, Geometrical mechanics (2 parts), Dept. of Math., Univ. of Chicago, Illinois, 1968.
  • 47. J. Marsden, Applications of global analysis in mathematical physics, Publish or Perish, Boston, 1974. MR 646816
  • 48. C. F. Martin and R. Hermann, Applications of algebraic geometry to systems theory. Part II: Feedback and pole placement for linear Hamiltonian systems, Proc. of the IEEE 65 (1977), 841-848. MR 527293
  • 49. C. F. Martin and R. Hermann, Applications of algebraic geometry to systems theory: The McMillan degree and Kronecker indices of transfer functions as topological and holomorphic system invariants, SIAM J. Control Optim. 16 (1978), 743-755. MR 527294
  • 50. D. Q. Mayne and R. W. Brockett (Editors), Geometric methods in systems theory, Reidel, Dordrecht, Holland, 1973.
  • 51. N. D. Mermin, The topological theory of defects in ordered media, Rev. Modern Phys. 51 (1979), no. 3, 591–648. MR 541885, https://doi.org/10.1103/RevModPhys.51.591
  • 52. C. Misner, K. Thorne and J. A. Wheeler, Gravitation, Freeman, San Francisco, 1973. MR 418833
  • 53. A. I. Murdoch and H. Cohen, Symmetry considerations for material surfaces, Arch. Rational Mech. Anal. 72 (1979/80), no. 1, 61–98. MR 540222, https://doi.org/10.1007/BF00250737
  • 54. W. Noll, Materially uniform simple bodies with inhomogeneitics, Arch. Rational Mech. Anal. 27(1967), 1-32. MR 225530
  • 55. B. N. Parlett and W. G. Poole, Jr., A geometric theory for the QR, LU and power iterations, SIAM J. Numer. Anal. 10 (1973), 389-412. MR 336979
  • 56. R. Penrose and R. S. Ward, Twistors for flat and curved space-time, General relativity and gravitation, Vol. 2, Plenum, New York-London, 1980, pp. 283–328. MR 617923
  • 57. V. Poénaru, Some aspects of the theory of defects of ordered media and gauge fields related to foliations, Comm. Math. Phys. 80 (1981), no. 1, 127–136. MR 623154
  • 58. R. Sachs and H. Wu, General relativity for mathematicians, Graduate Texts in Math., No. 48, Springer-Verlag, Berlin-Heidelberg and New York, 1977. MR 503498
  • 59. C. R. Schneider, Global aspects of the matrix Riccati equation, Math. Systems Theory 7 (1973), 281-286. MR 391175
  • 60. D. J. Simms and N. M. J. Woodhouse, Lectures on geometric quantization, Lecture Notes in Phys., No. 53, Springer-Verlag, Berlin-Heidelberg, 1976. MR 672639
  • 61. Jędrzej Śniatycki, Geometric quantization and quantum mechanics, Applied Mathematical Sciences, vol. 30, Springer-Verlag, New York-Berlin, 1980. MR 554085
  • 62. C. P. Snow, The search, 2nd ed., Schribner, New York, 1958, pp. 203-209.
  • 63. J. M. Souriau, Structure des systemes dynamiques, Dunod, Paris, 1970. MR 260238
  • 64. S. Sternberg, Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N. J., 1963. MR 193578
  • 65. S. Sternberg, Review of "Foundations of mechanics" by Ralph Abraham and Jerrold E. Marsden, Bull. Amer. Math. Soc. (N. S.) 2 (1980), 378-387.
  • 66. H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations 12 (1972), 95-116. MR 338882
  • 67. H. J. Sussmann, Existence and uniqueness of minimal realizations of nonlinear systems, Math. Systems Theory 10 (1977), 263-284. MR 437158
  • 68. Héctor J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations 31 (1979), no. 1, 31–52. MR 524816, https://doi.org/10.1016/0022-0396(79)90151-7
  • 69. Walter Thirring, A course in mathematical physics. Vol. I. Classical dynamical systems, Springer-Verlag, New York-Vienna, 1978. Translated from the German by Evans M. Harrell. MR 507189
  • 70. Walter Thirring, A course in mathematical physics. Vol. 2, Springer-Verlag, New York-Vienna, 1979. Classical field theory; Translated from the German by Evans M. Harrell. MR 553112
  • 71. Andrzej Trautman, Fiber bundles, gauge fields, and gravitation, General relativity and gravitation, Vol. 1, Plenum, New York-London, 1980, pp. 287–308. MR 583721
  • 72. N. R. Wallach, Symplectic geometry and Fourier analysis, Lie Groups: History, Frontiers and Applications, Vol. V, Math-Sci Press, Brookline, Mass., 1977. MR 488148
  • 73. C. -C. Wang, On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rational Mech. Anal. 27 (1967), 33-94. MR 229420
  • 74. L. P. Hughston and Richard Samuel Ward (eds.), Advances in twistor theory, Research Notes in Mathematics, vol. 37, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 578487
  • 75. R. O. Wells Jr., Complex manifolds and mathematical physics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 2, 296–336. MR 520077, https://doi.org/10.1090/S0273-0979-1979-14596-8
  • 76. A. Weinstein, Lectures on symplectic manifolds, CBMS Conf. Ser. in Math., No. 29, Amer. Math. Soc., Providence, R. I., 1977. MR 464312
  • 77. W. Murray Wonham, Linear multivariable control: a geometric approach, 2nd ed., Applications of Mathematics, vol. 10, Springer-Verlag, New York-Berlin, 1979. MR 569358

Review Information:

Reviewer: Clark R. Givens
Reviewer: Richard S. Millman
Journal: Bull. Amer. Math. Soc. 6 (1982), 467-478
DOI: https://doi.org/10.1090/S0273-0979-1982-15019-4
American Mathematical Society