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Symmetry aspects of nonholonomic field theories

Vankerschaver, Joris and Martín de Diego, David (2008) Symmetry aspects of nonholonomic field theories. Journal of Physics A: Mathematical and Theoretical, 41 (3). 035401. ISSN 1751-8113. http://resolver.caltech.edu/CaltechAUTHORS:VANjpa08

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Abstract

The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, describing the evolution of the associated component of the momentum map. Keeping up with the underlying geometric philosophy, a small modification of the derivation of the momentum lemma allows us to also treat generalized nonholonomic symmetries, which are vector fields along a projection. Such symmetries arise for example in practical examples of nonholonomic field theories such as the Cosserat rod, for which we recover both energy conservation (a previously known result) and a modified conservation law associated with spatial translations.


Item Type:Article
Additional Information:© 2008 IOP Publishing Limited. IOP Select Received 28 September 2007, in final form 28 November 2007. Published 4 January 2008. Print publication: Issue 3 (25 January 2008) The authors would like to thank F Cantrijn and D Saunders for stimulating discussions and comments. The first author is a Postdoctoral Fellow from the Research Foundation—Flanders (FWO-Vlaanderen), and a Fulbright Research Scholar at the California Institute of Technology. Additional financial support from the Fonds Professor Wuytack is gratefully acknowledged. The second author is supported by MEC (Spain) grants MTM 2004-7832 and MTM2007-62478, project ‘Ingenio Mathematica’ (i-MATH) no. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM.
Issue or Number:3
Record Number:CaltechAUTHORS:VANjpa08
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:VANjpa08
Alternative URL:http://dx.doi.org/10.1088/1751-8113/41/3/035401
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9521
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:29 Jan 2008
Last Modified:26 Dec 2012 09:49

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