On the canonical forms of the multi-dimensional averaged Poisson brackets

TL;DR

This paper investigates the canonical forms of multi-dimensional averaged Poisson brackets derived via the Whitham method, showing conditions under which they can be simplified to canonical or pseudo-canonical forms.

Contribution

It demonstrates that averaged brackets can be transformed into canonical or pseudo-canonical forms under specific conditions, extending understanding of their structure.

Findings

Averaged brackets can be canonical after Hydrodynamic Type transformations without annihilators.

In general, brackets can be transformed into a pseudo-canonical form under certain conditions.

The structure depends on the presence or absence of annihilators in the initial bracket.

Abstract

We consider here special Poisson brackets given by the "averaging" of local multi-dimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the "physical meaning" of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of "Hydrodynamic Type" in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a "pseudo-canonical" form under some special ("physical") requirements on the initial bracket.