Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

An infinite-dimensional Hamiltonian system on projective Hilbert space
HTML articles powered by AMS MathViewer

by Anthony M. Bloch PDF
Trans. Amer. Math. Soc. 302 (1987), 787-796 Request permission

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
    R. A. Abraham and J. E. Marsden, Foundations of mechanics, Benjamin/Cummings, 1978.
  • 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, DOI 10.1016/0001-8708(80)90008-0
  • V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
  • 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, DOI 10.1090/S0273-0979-1985-15365-0
  • —, Completely integrable Hamiltonian systems and total least squares estimation, Ph. D. Thesis, Harvard Univ., 1985.
  • Anthony Bloch, Estimation, principal components and Hamiltonian systems, Systems Control Lett. 6 (1985), no. 2, 103–108. MR 801020, DOI 10.1016/0167-6911(85)90005-2
  • —, 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).
  • 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, DOI 10.1016/0375-9601(86)90054-X
  • 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
  • C. I. Byrnes and J. C. Willems, Least squares estimation, linear programming and momentum, preprint.
  • Phillip A. Griffiths, Linearizing flows and a cohomological interpretation of Lax equations, Amer. J. Math. 107 (1985), no. 6, 1445–1484 (1986). MR 815768, DOI 10.2307/2374412
  • Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • D. G. Kendall, Multivariate analysis, Macmillan, 1975.
  • 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
  • Bernhard Maissen, Lie-Gruppen mit Banachräumen als Parameterräume, Acta Math. 108 (1962), 229–270 (German). MR 142693, DOI 10.1007/BF02545768
  • 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
  • A. S. Mischenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras, Select. Math. Soviet. 2 (1982), 207-292.
  • 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
  • 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, DOI 10.1007/978-0-8176-4578-6
  • Tudor Raţiu, The motion of the free $n$-dimensional rigid body, Indiana Univ. Math. J. 29 (1980), no. 4, 609–629. MR 578210, DOI 10.1512/iumj.1980.29.29046
  • 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
Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 302 (1987), 787-796
  • MSC: Primary 58F05; Secondary 58F07, 70H05, 81C99
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0891647-5
  • MathSciNet review: 891647