By Paul Newton, Phil Holmes, Alan Weinstein

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 quite short money owed of analysis and technical advances provided on the assembly, we wanted to recognize Jerry’s in?uence as a instructor, a propagator of recent rules, and a mentor of younger expertise. Con- quently, beginning in 1999, we sought to gather articles that will be used as access issues through scholars drawn to ?elds which were formed via Jerry’s paintings. while we was hoping to offer specialists engrossed of their personal technical niches a sign of the glorious breadth and intensity in their topics as an entire. This ebook is an end result of the e?orts of these who approved our in- tations to give a contribution. It offers either survey and learn 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 idea, g- metric mechanics, geometric keep an eye on idea, and relativity and quantum mechanics. the typical thread operating via this huge tapestry is using geometric equipment that serve to unify varied disciplines and produce a widevarietyofscientistsandmathematicianstogether,speakingalanguage which reinforces discussion and encourages cross-fertilization.

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The second approach is to consider the true minimizers of the discrete problems, and to try to understand what functional their limit minimizes. One example of this approach is the interesting study by Friesecke and Theil [2001] of a model two-dimensional problem of a lattice of particles linked by harmonic springs between their nearest and next nearest neighbours. They determine open regions of atomic parameters in which the Cauchy–Born hypothesis holds in the appropriate limit, and open regions in which it does not.

15) For the more general case of a thermoviscoelastic material (of strain-rate type), TR , η, ψ, qR are assumed to be functions of Dy, Dyt , θ, grad θ. By the same method we ﬁnd that ψ = ψ(Dy, θ), and that S · Dyt − η = −Dθ ψ , qR · grad θ ≥ 0, θ where TR = DA ψ + S Dy, Dyt , θ, grad θ . 15). 17) 1 for some matrix-valued function Σ, where U = (DyT Dy) 2 . 2 Existence of Solutions Problem 12. Prove the global existence and uniqueness of solutions to initial boundary-value problems for properly formulated dynamic theories of nonlinear elasticity.

For |C − 1| < γ we have that 1 W (CA) − W (A) = 0 d W (C(t)A) dt dt 1 DA W C(t)A · (C − 1)A dt = 0 1 = DA W C(t)A C(t)A T · (C − 1)C(t)−1 dt 0 1 ≤K W C(t)A + 1 · C − 1 · C(t)−1 dt 0 1 ≤ 2Kγ W C(t)A + 1 dt . 0 Let θ(A) = sup|C−1|<γ W (CA). 24) follows. 24) that DA W (CA)AT = DA W (CA)(CA)T C−T ≤ K W (CA) + 1 C−T ≤ 3K W (A) + 1 , as required. 4. (i) Given ϕ as in the theorem, deﬁne for |τ | sufﬁciently small yτ (x) := y(x) + τ ϕ y(x) . e. x ∈ Ω. 1. e. x ∈ Ω and limτ →0 yτ − y Hence I(yτ ) ≥ I(y) for |τ | suﬃciently small.