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Dissipation-induced instabilities in finite dimensions

Krechetnikov, R. and Marsden, J. E. (2007) Dissipation-induced instabilities in finite dimensions. Reviews of Modern Physics, 79 (2). pp. 519-553. ISSN 0034-6861. http://resolver.caltech.edu/CaltechAUTHORS:KRErmp07

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Abstract

The goal of this work is to introduce a coherent theory of the counterintuitive phenomena of dynamical destabilization under the action of dissipation. While the existence of one class of dissipation-induced instabilities was known to Sir Thomson (Lord Kelvin), it was not realized until recently that there is another major type of these phenomena hinted at by one of Merkin's theorems; in fact, these two cases exhaust all the generic possibilities. The theory grounded on the Thomson-Tait-Chetayev and Merkin theorems and on the geometric understanding introduced in this paper leads to the conclusion that ubiquitous dissipation is one of the paramount mechanisms by which instabilities develop in nature. Along with a historical review, the main theoretical achievements are put in a general context, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature. This general view also highlights the striking connection to various areas of mathematics. To appeal to the reader's intuition and experience, a large number of motivating examples are provided. The paper contains some new unpublished results and insights, and, finally, open questions are formulated to provide an impetus for future studies. While this review focuses on the finite-dimensional case, where the theory is relatively complete, a brief discussion of the current state of knowledge in the infinite-dimensional case, typified by partial differential equations, is also given.


Item Type:Article
Additional Information:©2007 The American Physical Society. (Published 4 April 2007)
Subject Keywords:classical mechanics; bifurcation; eigenvalues and eigenfunctions; elasticity; partial differential equations; deformation
Record Number:CaltechAUTHORS:KRErmp07
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:KRErmp07
Alternative URL:http://dx.doi.org/10.1103/RevModPhys.79.519
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10974
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:21 Jun 2008
Last Modified:26 Dec 2012 10:07

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