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Geometry, Mechanics, and Dynamics by John M. Ball (auth.), Paul Newton, Philip Holmes, Alan

By John M. Ball (auth.), Paul Newton, Philip Holmes, Alan Weinstein (eds.)

Jerry Marsden, one of many world’s pre-eminent mechanicians and utilized mathematicians, celebrated his sixtieth birthday in August 2002. the development used to be marked by means of a workshop on “Geometry, Mechanics, and Dynamics”at the Fields Institute for examine within the Mathematical Sciences, of which he wasthefoundingDirector. Ratherthanmerelyproduceaconventionalp- ceedings, with fairly short debts of analysis and technical advances offered on the assembly, we needed to recognize Jerry’s in?uence as a instructor, a propagator of latest principles, and a mentor of younger expertise. Con- quently, beginning in 1999, we sought to gather articles that would be used as access issues by means of scholars drawn to ?elds which have been formed by means of Jerry’s paintings. whilst we was hoping to provide specialists engrossed of their personal technical niches a sign of the glorious breadth and intensity in their topics as a complete. This publication is an final result of the e?orts of these who approved our in- tations to give a contribution. It offers either survey and examine articles within the a number of ?elds that symbolize the most topics of Jerry’s paintings, together with elasticity and research, ?uid mechanics, dynamical platforms conception, g- metric mechanics, geometric keep an eye on thought, and relativity and quantum mechanics. the typical thread working via this huge tapestry is using geometric tools that serve to unify varied disciplines and produce a widevarietyofscientistsandmathematicianstogether,speakingalanguage which boosts discussion and encourages cross-fertilization.

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In the context of free-energy minimization, one can draw a distinction between two kinds of approach. In the first, an appropriate limit of a discrete energy functional is sought along sequences of explicit atomic configurations. For example the atoms may be assumed to occupy a periodic lattice in a reference configuration, and to be displaced according to a given sufficiently smooth continuum deformation y (the Cauchy–Born hypothesis), the number of atoms being sent to infinity with a suitable scaling for the energy.

E(y) = Ω Since there are different types of local minimizer corresponding to different metrics d on different spaces X of deformations y, in particular W 1,∞ local minimizers and W 1,p local minimizers for 1 ≤ p < ∞, it is not clear which kinds of local minimizers are needed to ensure dynamical stability. As emphasised, for example, by Knops and Wilkes [1973], the standard argument for establishing Lyapunov stability with respect to a metric d requires more than just that y∗ is a strict local minimizer with respect to d (that is I(y) > I(y∗ ) whenever y = y∗ and d(y, y∗ ) is sufficiently small).

Given such a theory, the points at issue are the usual ones for dissipative dynamical systems, namely whether solutions converge to equilibrium states as t → ∞, the structure of regions of attraction, the existence of stable and unstable manifolds of equilibria, the existence of a global attractor, and so on. In particular one can ask whether dynamic orbits generically realize suitably defined local minimizing sequences for the ballistic free energy. 2). In fact for the one-dimensional viscoelastic model of this type studied by Ball, Holmes, James, Pego, and Swart [1991] and Friesecke and McLeod [1996], it is known that no dynamic solutions realize global minimizing sequences; it is unclear whether or not this is a one-dimensional phenomenon.

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