Abstract
It is shown that gradient equations on adjoint orbits with respect to certain metrics have multiple bracket representation. This is a generalization of Brockett’s dynamical system of double bracket formH = [H, [H, N]].
Similar content being viewed by others
References
A.M. Bloch, A completely integrable Hamiltonian system associated with line fitting in complex vector spaces. Bull. Amer. Math. Soc. (New series),23 (1985), 250–254.
A.M. Bloch, Steepest discent, linear programming and Hamiltonian Flows. Contemp. Math.,114 (1990), 77–88.
R.W. Brockett, Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems. Linear Algebra Appl.,146 (1991), 79–91.
R.W. Brockett, Differential geometry and the design of gradient algorithms. Differential Geometry: Partial Differential Equations on Manifolds (eds. R.E. Green and S.T. Yau), Amer. Math. Soc., Providence, 1993, 69–92.
P. Deift, T. Nanda, and C. Tomei, Differential equations for the symmetric eigenvalue problem. SIAM J. Numer. Anal.,20 (1983), 1–22.
H. Flaschka, The Toda lattice. II. Existence of integrals. Phys. Rev.,B9 (1974), 1924–1925.
U. Helmke, Isospectral flows on symmetric matrices and Riccati equation. System Control Lett.,16 (1991), 159–165.
J. Moser, Finitely many mass points on the line under the influence of an exponential potential-An integrable system. Dynamic Systems Theory and Applications (ed. J. Moser), Lecture Notes in Phys.,38, Springer-Verlag, Berlin-New York, 1975, 467–497.
Y. Nakamura, A new nonlinear dynamical system that leads to eigenvalues. Japan J. Indust. Appl. Math.,9 (1992), 133–139.
Y. Nakamura, Lax equations associated with a least squares problem and compact Lie algebras. Recent Developments in Differential Geometry (ed. K. Shiohama), Adv. Stud. Pure Math.,23, Math. Soc. Japan, 1993, 213–229.
M. Toda, Waves in nonlinear lattice. Progr. Theoret. Phys. Suppl.,45 (1970), 174–200.
Author information
Authors and Affiliations
About this article
Cite this article
Hori, G. Isospectral flows expressed in multiple bracket forms. Japan J. Indust. Appl. Math. 14, 315–327 (1997). https://doi.org/10.1007/BF03167387
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03167387