Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An infinite-dimensional Hamiltonian system on projective Hilbert space

Author: Anthony M. Bloch
Journal: Trans. Amer. Math. Soc. 302 (1987), 787-796
MSC: Primary 58F05; Secondary 58F07, 70H05, 81C99
MathSciNet review: 891647
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider here the explicit integration of a Hamiltonian system on infinite-dimensional complex projective space. The Hamiltonian, which is the restriction of a linear functional to this projective space, arises in the problem of line fitting in complex Hilbert space (or, equivalently, the problem of functional approximation) or as the expectation value of a model quantum mechanical system. We formulate the system here as a Lax system with parameter, showing how this leads to an infinite set of conserved integrals associated with the problem and to an explicit formulation of the flow in action-angle form via an extension of some work of J. Moser. In addition, we find the algebraic curve naturally associated with the system.

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

  • [1] R. A. Abraham and J. E. Marsden, Foundations of mechanics, Benjamin/Cummings, 1978.
  • [2] Mark Adler and Pierre van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. in Math. 38 (1980), no. 3, 318–379. MR 597730, 10.1016/0001-8708(80)90008-0
  • [3] 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
  • [4] Anthony M. Bloch, A completely integrable Hamiltonian system associated with line fitting in complex vector spaces, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 250–254. MR 776479, 10.1090/S0273-0979-1985-15365-0
  • [5] -, Completely integrable Hamiltonian systems and total least squares estimation, Ph. D. Thesis, Harvard Univ., 1985.
  • [6] Anthony Bloch, Estimation, principal components and Hamiltonian systems, Systems Control Lett. 6 (1985), no. 2, 103–108. MR 801020, 10.1016/0167-6911(85)90005-2
  • [7] -, Total least squares estimation in infinite dimensions and completely integrable Hamiltonian systems, Proc. 7th Internat. Conf. on the Mathematical Theory of Networks and Systems, Stockholm, 1985 (to appear).
  • [8] Anthony M. Bloch, An infinite-dimensional classical integrable system and the Heisenberg and Schrödinger representations, Phys. Lett. A 116 (1986), no. 8, 353–355. MR 850214, 10.1016/0375-9601(86)90054-X
  • [9] Anthony M. Bloch and Christopher I. Byrnes, An infinite-dimensional variational problem arising in estimation theory, Algebraic and geometric methods in nonlinear control theory, Math. Appl., vol. 29, Reidel, Dordrecht, 1986, pp. 487–498. MR 862339
  • [10] C. I. Byrnes and J. C. Willems, Least squares estimation, linear programming and momentum, preprint.
  • [11] Phillip A. Griffiths, Linearizing flows and a cohomological interpretation of Lax equations, Amer. J. Math. 107 (1985), no. 6, 1445–1484 (1986). MR 815768, 10.2307/2374412
  • [12] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [13] D. G. Kendall, Multivariate analysis, Macmillan, 1975.
  • [14] Pierre de la Harpe, Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space, Lecture Notes in Mathematics, Vol. 285, Springer-Verlag, Berlin-New York, 1972. MR 0476820
  • [15] Bernhard Maissen, Lie-Gruppen mit Banachräumen als Parameterräume, Acta Math. 108 (1962), 229–270 (German). MR 0142693
  • [16] H. P. McKean, Integrable systems and algebraic curves, Global analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ. Calgary, Calgary, Alta., 1978) Lecture Notes in Math., vol. 755, Springer, Berlin, 1979, pp. 83–200. MR 564904
  • [17] A. S. Mischenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras, Select. Math. Soviet. 2 (1982), 207-292.
  • [18] J. Moser, Geometry of quadrics and spectral theory, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 147–188. MR 609560
  • [19] David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776
  • [20] Tudor Raţiu, The motion of the free 𝑛-dimensional rigid body, Indiana Univ. Math. J. 29 (1980), no. 4, 609–629. MR 578210, 10.1512/iumj.1980.29.29046
  • [21] Satosi Watanabe, Karhunen-Loève expansion and factor analysis: Theoretical remarks and applications, Trans. Fourth Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes (Prague, 1965) Academia, Prague, 1967, pp. 635–660. MR 0234768

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F05, 58F07, 70H05, 81C99

Retrieve articles in all journals with MSC: 58F05, 58F07, 70H05, 81C99

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society