Nonlocal orientation-dependent dynamics of charged strands and ribbons

Holm, Darryl D.; Putkaradze, Vakhtang

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

Nous établir les équations de la dynamique hamiltonienne d'une courbe (chaine moléculaire) dans l'espace physique ${\mathbb{R}}^3$ 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 $\mathit{so}(3)Ⓢ({\mathbb{R}}^3\oplus {\mathbb{R}}^3\oplus {\mathbb{R}}^3\oplus {\mathbb{R}}^3)$ (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 $\mathit{SO}(3)$ sur son algébre $\mathit{so}(3)$. Les nouvelles équations du filament sont écrites sous une forme conservative grâce aux actions coadjointes correspondantes.

Time-dependent Hamiltonian dynamics is derived for a strand of charged units in ${\mathbb{R}}^3$ 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 $\mathit{so}(3)Ⓢ({\mathbb{R}}^3\oplus {\mathbb{R}}^3\oplus {\mathbb{R}}^3\oplus {\mathbb{R}}^3)$ 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 $\mathit{SO}(3)$ on its Lie algebra $\mathit{so}(3)$. The new strand equations are written in conservative form by using the corresponding coadjoint actions.

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