Abstract
In this paper, we calculate all possible null Lagrangians (null energies) for the mechanics of a distinguished class of continua, the nematic elastomers. The calculation is done in order to help to relate different physically equivalent theories of nematic elastomers. We discuss both local and global (hence topological) aspects of the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. M. Anderson and T. Duchamp, “On the existence of global variational principles,” Amer. Math. J., 102, 781–868 (1980).
D. R. Anderson, D. E. Carlson, E. Fried, “A continuum-mechanical theory for nematic elastomers,” J. Elasticity, 56, 35–58 (1999).
A. V. Bocharov et al., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (I. S. Krasil’shchik and A. M. Vinogradov, Eds.), Amer. Math. Soc. (1999).
R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Grad. Texts Math., 82, Springer-Verlag, Berlin (1982).
J. L. Ericksen, “Conservation laws for liquid crystals,” Trans. Soc. Rheol., 5, 23–34 (1961).
J. L. Ericksen, “Nilpotent energies in liquid crystal theories,” Arch. Rational Mech. Anal., 10, 189–196 (1962).
E. Fried and S. Sellers, “Free energy-density functions for nematic elastomers,” J. Mech. Phys. Solids, 52, 1671–1689 (2004).
D. R. Grigore, “Variationally trivial Lagrangians and locally variational differential equations of arbitrary order,” Differ. Geom. Appl., 10, 79–105 (1999).
D. Krupka, “Variational sequences on finite-order jet spaces,” in: Proc. Conf. on Differential Geometry and Its Appl., World Scientific, New York (1990), pp. 236–254.
D. Krupka and J. Musilova, “Trivial Lagrangians in field theory,” Differ. Geom. Appl., 9, 293–505 (1998).
F. M. Leslie, “Some constitutive equations for liquid crystals,” Arch. Rational Mech. Anal., 28, 265–283 (1968).
F. M. Leslie, I. W. Stewart, T. Carlsson and M. Nakagawa, “Equivalent smectic C liquid crystal energies,” Contin. Mech. Thermodyn., 3, 237–250 (1991).
P. J. Olver, Applications of Lie Groups to Differential Equations, Grad. Texts Math., 107, Springer-Verlag (1986).
P. J. Olver and J. Sivaloganathan, “The structure of null Lagrangians,” Nonlinearity, 1, 389–398 (1988).
M. Palese, R. Vitolo, E. Winterroth, “Minimal order problems in the calculus of variations,” in preparation.
D. J. Saunders, The Geometry of Jet Bundles, Cambridge Univ. Press (1989).
E. M. Terentjev, “Liquid-cristalline elastomers,” J. Phys. Condensed Matter, 11, R239–R257 (1999).
W. M. Tulczyjew, “The Lagrange complex,” Bull. Soc. Math. France, 105, 419–431 (1977).
A. M. Vinogradov, “The C-spectral sequence, Lagrangian formalism and conservation laws, I, II,” J. Math. Anal. Appl., 100 (1984).
E. Virga, Variational Theories for Liquid Crystals, Appl. Math. and Math. Comput., 8, Chapman & Hall (1994).
R. Vitolo, “On different geometric formulations of Lagrangian formalism,” Differ. Geom. Appl., 10, 225–255 (1999).
R. Vitolo, “The finite-order C-spectral sequence,” Acta Appl. Math., 72, 133–154 (2002).
M. Warner and E. M. Terentjev, “Nematic elastomers—a new state of matter?” Progr. Polym. Sci., 21, 853–891 (1996).
Author information
Authors and Affiliations
Additional information
__________
Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.
Rights and permissions
About this article
Cite this article
Saccomandi, G., Vitolo, R. Null Lagrangians for nematic elastomers. J Math Sci 136, 4470–4477 (2006). https://doi.org/10.1007/s10958-006-0238-z
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0238-z