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Classification of multidimensional Poisson brackets of hydrodynamic type. (English. Russian original) Zbl 1123.37029

Russ. Math. Surv. 61, No. 2, 356-358 (2006); translation from Usp. Mat. Nauk 61, No. 2, 167-168 (2006).
Let \(u-(u^1,\dots,u^N)\) be local coordinates in a smooth \(N\)-dimensional manifold \(M\), \(x=(x^1,\dots,x^n)\) and \(y=(y^1,\dots, y^n)\) be independent variables, \(g^{ij\alpha}(u)\) and \(b_k^{ij\alpha} (u)\) be smooth functions, \(u(x)=(u^1(x),\dots,u^N(x))\) be smooth functions with values in \(M\). The author recalls the Poisson bracket of hydrodynamical type \[ \{u^i(x),u^j(y)\}=\sum (g^{ij\alpha}(u(x))\frac {\partial\delta}{\partial x^\alpha}(x-y)+b_k^{ij\alpha} (u(x))\frac {\partial u^k(x)}{\partial x^\alpha}\delta(x-y)) \] and gives a short report on his recent solution of the classification problem in the nondegenerate case. The methods of differential geometry are applied: \(g^{ij\alpha}\) represent flat Riemannian metrics and the formulae \(b_k^{ij\alpha}=-\sum g^{is\alpha} \Gamma^{j\alpha}_{sk}\) provide the link to the Levi-Civita connections. The results are expressed in terms of the tensors \(T^i_{jk}=\Gamma^{i\alpha}_{jk}-\Gamma_{jk}^{i \beta}\).

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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