Journal of Geometry and Symmetry in Physics

Multidimentional Poisson Brackets of Hydrodynamic Type and Flat Pencils of Metrics

Oleg I. Mokhov

Abstract

We give an exposition of some recent crucial achievements in the theory of multidimensional Poisson brackets of hydrodynamic type. In particular, we solve the well-known Dubrovin–Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. We obtain the classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number $N$ of components and for any dimension $n$ by differential-geometric methods. This problem is equivalent to the classification of a special class of flat pencils of metrics. A key role in the solution of this problem was played by the theory of compatible metrics that had been earlier constructed by the present author.

Article information

Source
J. Geom. Symmetry Phys., Volume 10 (2007), 51-72.

Dates
First available in Project Euclid: 20 May 2017

https://projecteuclid.org/euclid.jgsp/1495245623

Digital Object Identifier
doi:10.7546/jgsp-10-2007-51-72

Mathematical Reviews number (MathSciNet)
MR2380050

Zentralblatt MATH identifier
1170.37027

Citation

Mokhov, Oleg I. Multidimentional Poisson Brackets of Hydrodynamic Type and Flat Pencils of Metrics. J. Geom. Symmetry Phys. 10 (2007), 51--72. doi:10.7546/jgsp-10-2007-51-72. https://projecteuclid.org/euclid.jgsp/1495245623