Nonlocal orientation-dependent dynamics of charged strands and ribbons

Holm, Darryl D.; Putkaradze, Vakhtang

doi:10.1016/j.crma.2009.06.009

Mathematical Physics/Mathematical Problems in Mechanics

Nonlocal orientation-dependent dynamics of charged strands and ribbons

Presented by: Charles-Michel Marle

Holm, Darryl D.¹; Putkaradze, Vakhtang ^2,³

¹ Department of Mathematics, Imperial College London, London SW7 2AZ, UK
² Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
³ Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA

Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1093-1098.

Abstract
Résumé

Time-dependent Hamiltonian dynamics is derived for a strand of charged units in $R^{3}$ held together by both nonlocal (for example, electrostatic) and elastic interactions. The dynamical equations in the symmetry-reduced variables are written on the dual of the semidirect-product Lie algebra $so (3) Ⓢ (R^{3} \oplus R^{3} \oplus R^{3} \oplus R^{3})$ with three 2-cocycles. We also demonstrate that the nonlocal interaction produces an interesting new term deriving from the coadjoint action of the Lie group $SO (3)$ on its Lie algebra $so (3)$ . The new strand equations are written in conservative form by using the corresponding coadjoint actions.

Nous établir les équations de la dynamique hamiltonienne d'une courbe (chaine moléculaire) dans l'espace physique $R^{3}$ sujette á des interactions élastiques ainsi que non-locales (électrostatiques par exemple). Les équations dynamiques des variables réduites par symétrie sont écrites sur l'espace dual de l'algèbre de Lie $so (3) Ⓢ (R^{3} \oplus R^{3} \oplus R^{3} \oplus R^{3})$ (produit semidirect) avec trois 2-cocycles. Nous démontrons aussi que l'interaction non-locale produit un nouvel terme intéressant, qui dérive de l'action coadjointe du group de Lie $SO (3)$ sur son algébre $so (3)$ . Les nouvelles équations du filament sont écrites sous une forme conservative grâce aux actions coadjointes correspondantes.

Received: 2008-04-23
Accepted: 2009-06-05
Published online: 2009-07-11

DOI: 10.1016/j.crma.2009.06.009

Author's affiliations:

Holm, Darryl D. ¹; Putkaradze, Vakhtang ^{2, 3}

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     title = {Nonlocal orientation-dependent dynamics of charged strands and ribbons},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1093--1098},
     publisher = {Elsevier},
     volume = {347},
     number = {17-18},
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     doi = {10.1016/j.crma.2009.06.009},
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     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.06.009/}
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Holm, Darryl D.; Putkaradze, Vakhtang. Nonlocal orientation-dependent dynamics of charged strands and ribbons. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1093-1098. doi : 10.1016/j.crma.2009.06.009. http://www.numdam.org/articles/10.1016/j.crma.2009.06.009/

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