Deformierbares Objekt , Riemannscher Raum , Fréchet-Mannigfaltigkeit , Noether-Theorem
Abstract:
The quality of a smoothly deformable medium in a Riemannian manifold N is characterized by a smooth one form F on the space of configurations, a Fréchet manifold. F charaeterizes the work done at a configuration under an infinitesimal distortion. In fact F determines a smooth constitutive vector field H on that Fréchet manifold. A symplectic setting is deduced to determine the equation of motion of the medium subjected to F. The equation - not hamiltonian in general - is such that ∇ d\dt σ( t) balances along the curve of configurations σ; the (internal) force density σ (σ(t)) H(σ(t)) with σ(σ(t)) the Laplacian determined by σ(t) and ∇ the covariant derivative on the space of configurations. We exhibit the effect to this equation of the work done by distorting the volume. No balance laws are presupposed in deriving the equation. Two types of such laws are derived from the symplectic setting on one hand and on the other from the Noether theorem, provided symmetry groups are present.
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