By Y. Gan

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The passage from χq to τq is exactly the one done by E. and F. Cosserat in ([C], p 190). 3. 4. For any section f q+1 ∈ Rq+1 , the transformation: ¯ f q +1 χq −→ χq = f q−+11 ◦ χq ◦ j1 ( f q ) + D ¯ χq = 0. is called a gauge transformation and exchanges the solutions of the ﬁeld equations D Introducing the formal Lie derivative on Jq ( T ) by the formulas: L(ξ q+1 )ηq = {ξ q+1 , ηq+1 } + i (ξ ) Dηq+1 = [ξ q , ηq ] + i (η ) Dξ q+1 ( L( j1 (ξ q+1 ))χq )(ζ ) = L(ξ q+1 )(χq (ζ )) − χq ([ξ, ζ ]) and passing to the limit with f q+1 = idq+1 + tξ q+1 + ...

Subsequently, Pantano et al. (2004) adopted a continuum approach and finite element method to investigate the morphology of the collapsed CNTs. In succession, Tang et al. studied the collapse of nanotubes using an inextensible elastica model. In use of phase plane analysis, they showed that CNTs can take collapsed configurations of different orders (Tang and Glassmaker, 2010). , 2005b). This significant difference from the result of Gao et al. is because a distinct force field and physical parameters were selected, which greatly affect the results of analytical solutions and molecular simulations.

Appell: Traité de Mécanique Rationnelle, Gauthier-Villars, Paris, 1909. Particularly t II concerned with analytical mechanics and t III with a Note by E. and F. Cosserat "Note sur la théorie de l’action Euclidienne", 557-629. [2] V. Arnold: Méthodes mathématiques de la mécanique classique, Appendice 2 (Géodésiques des métriques invariantes à gauche sur des groupes de Lie et hydrodynamique des ﬂuides parfaits), MIR, moscow, 1974,1976. -F. POMMARET: Arnold’s hydrodynamics revisited, AJSE-mathematics, 1, 1, 2009, 157-174).