Some applications of geometry in continuum mechanics

Hughes, T. J. R. and J. E. Marsden

Rep. Math. Phys., 12, (1977), 35-44

Abstract:

Some contemporary ideas from differential geometry are applied to continuum mechanics. The Lie derivative is used to clarify the notion of "objective rates", an intrinsic treatment of Piola transformations is described, a simplified proof of Vainberg's theorem for potential operators is given by way of the Poincaré lemma on infinite dimensional manifolds, and a new derivation of the basic equations of continuum mechanics is presented which is valid in a general Riemannian manifold setting.