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The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates. (English) Zbl 0668.73014

The authors systematically and explicitly derive the Hamiltonian structure of nonlinear three-dimensional elastodynamics, of rigid body dynamics and of geometrically exact rod and plate dynamical models. Each of these four topics is preceded by an introductory section to the corresponding theory in the geometric (covariant) spirit of the second author and T. J. R. Hughes [Mathematical foundations of elasticity (1983; Zbl 0545.73031)]. In each instance, three representations are studied: the material (Lagrangian) representation, the spatial (Eulerian) representation and the convected representation which is essentially a pull-back of the spatial representation onto the reference configuration. These introductory sections contain a wealth of useful formulae pertaining to each subject. The main section for each of the four topics consists in a derivation of the Hamiltonian form of the equations of motion in the Poisson bracket formalism $\stackrel{˙}{f}=\left\{f,H\right\}$. Canonical and reduced Poisson brackets are systematically computed in each case and for each representation.

The article can thus be used as a reference source for developments based on the Hamiltonian structure of elastodynamics, e.g., among those announced here, nonlinear stability of flexible structures or numerical schemes that exactly preserve important physical quantities.

Reviewer: H.Le Dret
##### MSC:
 74B20 Nonlinear elasticity 70H05 Hamilton’s equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics 74K20 Plates (solid mechanics) 70E15 Free motion of a rigid body