×

Hamiltonian pairs associated with skew-symmetric Killing tensors on spaces of constant curvature. (English. Russian original) Zbl 0832.53027

Funct. Anal. Appl. 28, No. 2, 123-125 (1994); translation from Funkts. Anal. Prilozh. 28, No. 2, 60-63 (1994).
Let \(u = \{u^1(x), \dots, u^n(x)\}\) denote the vector functions and \(F = \int f(u, u_x, \dots) dx\), \(G=\int g(u, u_x, \dots) dx\) the functionals. The authors investigate the conditions for the Poisson brackets \[ \{F,G\} = \int {\delta F\over \delta u^i (x)} \Omega^{ij} {\delta G\over \delta u^j (x)} dx, \] where \(\Omega^{ij}\) is equal to either \(A^{ij} = g^{ij}(u)d - g^{is} (u) \Gamma^j_{sk} (u) u^k_x + cu^i_x d^{-1} u^j_x\) or to \(\omega^{ij} (u)\), which is a skew-symmetric tensor, to be compatible. In other words, \(A^{ij}\), \(\omega^{ij}\) form a Hamiltonian pair, i.e. \(\omega^{ij}\) is a Killing tensor. As an application they show that the corresponding hierarchy of integrable systems coincides with the Heisenberg ferromagnet equations and construct the proper Killing tensors on the \(n\)-dimensional sphere.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] O. I. Mokhov and E. V. Ferapontov, Usp. Mat. Nauk,45, No. 3, 191-192 (1990).
[2] B. A. Dubrovin and S. P. Novikov, Dokl. Akad. Nauk SSSR,270, No. 4, 781-785 (1983).
[3] O. I. Mokhov, Phys. Letters A,166, No. 3, 4, 215-216 (1992).
[4] E. V. Ferapontov, Funkts. Anal. Prilozhen.,25, No. 3, 37-49 (1991).
[5] E. V. Ferapontov, Teor. Mat. Fiz.,91, No. 3, 452-462 (1992).
[6] E. V. Ferapontov, Funkts. Anal. Prilozhen.,26, No. 4, 83-86 (1992).
[7] V. E. Zakharov and L. A. Takhtadzhyan, Teor. Mat. Fiz.,38, No. 1, 26-35 (1979).
[8] Yu. N. Sidorenko, Zap. Nauchn. Sem. LOMI,161, No. 7, 76-87 (1987).
[9] E. Barouch, A. S. Fokas, and V. G. Papageorgiou, J. Math. Phys.,29, No. 12, 2628-2633 (1988). · Zbl 0783.58033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.