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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
96/2005

Metriplectic Structure, Leibniz Dynamics and Dissipative Systems

Partha Guha

Abstract

A metriplectic (or Leibniz) structure on a smooth manifold is a pair of skew-symmetric Poisson tensor $P$ and symmetric metric tensor $G$. The dynamical system defined by the metriplectic structure can be expressed in terms of Leibniz bracket. This structure is used to model the geometry of the dissipative systems. The dynamics of purely dissipative systems are defined by the geometry induced on a phase space via a metric tensor. The notion of Leibniz brackets is extendable to infinite-dimensional spaces. We study metriplectic structure compatible with the Euler-Poincaré framework of the Burgers and Whitham-Burgers equations. This means metricplectic structure can be constructed via Euler-Poincaré formalism. We also study the Euler-Poincaré frame work of the Holm-Staley equation, and this exhibits different type of metriplectic structure. Finally we study the 2D Navier-Stokes using metriplectic techniques.

Received:
Nov 1, 2005
Published:
Nov 1, 2005
MSC Codes:
58D05, 35Q5
Keywords:
metriplectic, leibniz bracket, burgers equation

Related publications

inJournal
2007 Repository Open Access
Partha Guha

Metriplectic structure, Leibniz dynamics and dissipative systems

In: Journal of mathematical analysis and applications, 326 (2007) 1, pp. 121-136