Comptes Rendus
no. 17-18
Mathematical Physics/Mathematical Problems in Mechanics
Nonlocal orientation-dependent dynamics of charged strands and ribbons
[Dynamique non-locale des filaments électisés]

Présenté par : Charles-Michel Marle

Darryl D. Holm1 ; Vakhtang Putkaradze2, 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.

Nous établir les équations de la dynamique hamiltonienne d'une courbe (chaine moléculaire) dans l'espace physique R3 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)Ⓢ(R3⊕R3⊕R3⊕R3) (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.

Time-dependent Hamiltonian dynamics is derived for a strand of charged units in R3 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)Ⓢ(R3⊕R3⊕R3⊕R3) 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.06.009
Darryl D. Holm 1 ; Vakhtang Putkaradze 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 
@article{CRMATH_2009__347_17-18_1093_0,
     author = {Darryl D. Holm and Vakhtang Putkaradze},
     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},
     year = {2009},
     doi = {10.1016/j.crma.2009.06.009},
     language = {en},
}
TY  - JOUR
AU  - Darryl D. Holm
AU  - Vakhtang Putkaradze
TI  - Nonlocal orientation-dependent dynamics of charged strands and ribbons
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 1093
EP  - 1098
VL  - 347
IS  - 17-18
PB  - Elsevier
DO  - 10.1016/j.crma.2009.06.009
LA  - en
ID  - CRMATH_2009__347_17-18_1093_0
ER  - 
%0 Journal Article
%A Darryl D. Holm
%A Vakhtang Putkaradze
%T Nonlocal orientation-dependent dynamics of charged strands and ribbons
%J Comptes Rendus. Mathématique
%D 2009
%P 1093-1098
%V 347
%N 17-18
%I Elsevier
%R 10.1016/j.crma.2009.06.009
%G en
%F CRMATH_2009__347_17-18_1093_0
Darryl D. Holm; Vakhtang Putkaradze. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.009/

[1] A. Balaeff; L. Mahadevan; K. Schulten Elastic rod model of a DNA loop in the lac operon, Phys. Rev. Lett., Volume 83 (1999), pp. 4900-4903

[2] J.R. Banavar; T.X. Hoang; J.H. Maddocks; A. Maritan; C. Poletto; A. Stasiak; A. Trovato Structural motifs of macromolecules, Proc. Natl. Acad. Sci., Volume 104 (2007), pp. 17283-17286

[3] T.C. Bishop; R. Cortez; O.O. Zhmudsky Investigation of bend and shear waves in a geometrically exact elastic rod model, J. Comp. Phys., Volume 193 (2004), pp. 642-665

[4] H. Cendra; J.E. Marsden Geometric mechanics and the dynamics of asteroid pairs, Dynamical Systems, Volume 20 (2005), pp. 3-21

[5] H. Cendra, J.E. Marsden, T.S. Ratiu, Lagrangian Reduction by Stages, Memoirs American Mathematical Society, vol. 152, 2001

[6] N. Chouaieb; A. Goriely; J.H. Maddocks Helices, Proc. Natl. Acad. Sci., Volume 103 (2006), pp. 9398-9403

[7] D. Dichmann; Y. Li; J.H. Maddocks Hamiltonian formulations and symmetries in rod mechanics, Minneapolis, MN, 1994 (IMA Vol. Math. Appl.), Volume vol. 82, Springer, New York (1996), pp. 71-113

[8] F. Gay-Balmaz; T.S. Ratiu The geometric structure of complex fluids, Adv. Appl. Math., Volume 42 (2009) no. 2, pp. 176-275

[9] J. Gibbons; D.D. Holm; B.A. Kupershmidt Gauge-invariant Poisson brackets for chromohydrodynamics, Phys. Lett. A, Volume 90 (1982), pp. 281-283

[10] R. Goldstein; A. Goriely; G. Huber; C. Wolgemuth Bistable helixes, Phys. Rev. Lett., Volume 84 (2000), pp. 1631-1634

[11] R. Goldstein; T.R. Powers; C.H. Wiggins Viscous nonlinear dynamics of twist and writhe, Phys. Rev. Lett., Volume 80 (1998), pp. 5232-5235

[12] A. Hausrath; A. Goriely Repeat protein architectures predicted by a continuum representation of fold space, Protein Sci., Volume 15 (2006), pp. 1-8

[13] D.D. Holm Euler–Poincaré dynamics of perfect complex fluids, Geometry, Mechanics and Dynamics, Special Volume in Honor of J.E. Marsden (2001), pp. 113-167

[14] D.D. Holm; B.A. Kupershmidt The analogy between spin glasses and Yang–Mills fluids, J. Math. Phys., Volume 29 (1988), pp. 21-30

[15] I. Mezic On the dynamics of molecular conformation, Proc. Natl. Acad. Sci., Volume 103 (2006), pp. 7542-7547

[16] J.C. Simó; J.E. Marsden; P.S. Krishnaprasad The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates, Arch. Rational Mech. Anal., Volume 104 (1988), pp. 125-183

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Geometric dissipation in kinetic equations

Darryl D. Holm; Vakhtang Putkaradze; Cesare Tronci

C. R. Math (2007)

Exact geometric theory for flexible, fluid-conducting tubes

François Gay-Balmaz; Vakhtang Putkaradze

C. R. Méca (2014)