The Hamiltonian structure of continuum mechanics in material, inverse material, spatial and convective representations. (English) Zbl 0611.70015
Hamiltonian structure and Lyapunov stability for ideal continuum dynamics, Sémin. Math. Supér., Sémin. Sci. OTAN (NATO Adv. Study Inst.) 100, 11-124 (1986).
[For the entire collection see Zbl 0594.00020.]
The paper is concerned with the theory of Hamiltonian structures in ideal continuum models. One main goal is to relate material and spatial representations of continuum mechanics to inverse material and convective representations, each of which admits symplectic structure.
The authors develop the abstract picture of reduction on dual bundles and treat Hamiltonian systems that transform as semidirect products under the action of the symmetry group of the configuration space.
Unifying preceding works on this area the authors define abstractly the above named continuum representations and maps between them, generated by groups and an inversion operator. Main examples illustrating these representations are free boundary problems for incompressible fluids, the heavy top, and the ideal, compressible, adiabatic fluid.
The paper is concerned with the theory of Hamiltonian structures in ideal continuum models. One main goal is to relate material and spatial representations of continuum mechanics to inverse material and convective representations, each of which admits symplectic structure.
The authors develop the abstract picture of reduction on dual bundles and treat Hamiltonian systems that transform as semidirect products under the action of the symmetry group of the configuration space.
Unifying preceding works on this area the authors define abstractly the above named continuum representations and maps between them, generated by groups and an inversion operator. Main examples illustrating these representations are free boundary problems for incompressible fluids, the heavy top, and the ideal, compressible, adiabatic fluid.
Reviewer: G.P.Ostermeyer
MSC:
| 70H05 | Hamilton’s equations |
| 55R25 | Sphere bundles and vector bundles in algebraic topology |
| 57S25 | Groups acting on specific manifolds |
| 70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |
| 57R50 | Differential topological aspects of diffeomorphisms |
| 76A02 | Foundations of fluid mechanics |
| 74Axx | Generalities, axiomatics, foundations of continuum mechanics of solids |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
