Tsarëv, S. P. On the Liouville Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type arising in the Bogolyubov-Whitham averaging theory. (English. Russian original) Zbl 0582.58012 Russ. Math. Surv. 39, No. 6, 227-228 (1984); translation from Usp. Mat. Nauk 39, No. 6(240), 209-210 (1984). The author considers one-dimensional Hamiltonian systems of hydrodynamic type arising in the Bogolyubov-Whitham averaging theory. He solves the problem of the classification of Liouville coordinate systems arising in averaging by the Bogolyubov-Whitham method [see B. A. Dulvovin and S. P. Novikov, Dokl. Akad. Nauk SSSR 270, 781-785 (1983; Zbl 0553.35011)], and their relation with diagonalizability (i.e. integrability) of the resulting system. Reviewer: N.Jacob Cited in 2 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37N99 Applications of dynamical systems Keywords:Hamiltonian systems of hydrodynamic type; Bogolyubov-Whitham averaging theory; Liouville coordinate Citations:Zbl 0553.35011 PDFBibTeX XMLCite \textit{S. P. Tsarëv}, Russ. Math. Surv. 39, No. 6, 227--228 (1984; Zbl 0582.58012); translation from Usp. Mat. Nauk 39, No. 6(240), 209--210 (1984) Full Text: DOI