Abstract
We show how a certain class of Hamiltonian systems give rise to differential equations on spaces of matrices whose elements are rational functions. In particular, we reinterpret the results of Kac and van Moerbeke on the periodic Toda lattice in terms of such differential equations and relate the action-angle coordinates found by them to the evolution of a certain two by two symmetric matrix of rational functions flowing on a space of fixed McMillian degree and fixed Cauchy index. Realization theory is used to pass from a description of the flow in terms of rational matrices to a description in terms of the original coordinates.
This work was supported in part by the National Science Foundation under Engineering Research Center Program, NSF EEC 94-02384, the US Army Research Office under grants DAAL03-92-G-0115 and DAAG55-97-1-0114.
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References
Paul Fuhrmann, Linear Systems and Operators in Hilbert Space, McGraw-Hill, New York, 1981.
M. Kac and P. van Moerbeke, “A complete solution to the periodic Toda problem,” PNAS, vol. 72, pp. 2879–2880, 1976
J. Moser, “Finitely many points on the line under the influence of an exponential potential-an integrable system,” in Dynamical Systems, Theory and Applications, (J. Moser, Ed.), Lecture notes in Physics, Vol. 38, Springer-Verlag, 1975.
P. S. Krishnaprasad, “Symplectic mechanics and Rational Functions,” Ricerche di Automatica, Vol. 10, (1979), pp. 107–135.
R. W. Brockett and Leonid Faybusovich, “Toda Flows, Inverse Spectral Transform and Realization Theory,” Systems and Control Letters, Vol. 16, No. 2 (1991) pp. 79–88.
A. M. Bloch, R. W. Brockett and T. Ratiu, “A New Formulation of the Generalized Toda Lattice Equations and their Fixed Point Analysis via the Moment Map,” Bulletin of the American Mathematical Society, Vol. 23, No 2 (1990) pp. 477–485.
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Dedicated to my long-time friend and colleague, Paul Fuhrmann, on the occasion of his 60th birthday
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© 1997 Springer Fachmedien Wiesbaden
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Brockett, R.W. (1997). A Rational Flow for the Toda Lattice Equations. In: Helmke, U., Prätzel-Wolters, D., Zerz, E. (eds) Operators, Systems and Linear Algebra. European Consortium for Mathematics in Industry. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-09823-2_4
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DOI: https://doi.org/10.1007/978-3-663-09823-2_4
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-663-09824-9
Online ISBN: 978-3-663-09823-2
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