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Relativistic elastodynamics
General Relativity and Quantum Cosmology · Submitted 4 May 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This way we can show well-posedness for the equations describing elastic matter on a fixed gravitational background, self-gravitating elastic matter and a finite elastic body on a fixed background.
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The abstract, sentence by sentence
There are many different formulations of relativistic elasticity. Most of them are only concerned with formal questions rather than questions regarding the PDE point of view. The aim of this thesis is to obtain various local existence results for dynamical scenarios involving elastic matter. Following Beig and Schmidt we treat relativistic elasticity as a Lagrangian field theory. The basic unknowns are mappings between spacetime and an abstract material manifold. The equations of motion will be the Euler-Lagrange equations arising from the action principle. Under certain (physically reasonable) restrictions on the elastic properties of the material the resulting system is hyperbolic and local existence results can be derived. This way we can show well-posedness for the equations describing elastic matter on a fixed gravitational background, self-gravitating elastic matter and a finite elastic body on a fixed background. The thesis also contains some results on energy conservation and on linearized elasticity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | There are many different formulations of relativistic elasticity. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s1 | Most of them are only concerned with formal questions rather than questions regarding the PDE point of view. | background 0.57 | 0.06 | 0.02 | asserted 0.99 |
| s2 | The aim of this thesis is to obtain various local existence results for dynamical scenarios involving elastic matter. | contribution 0.99 | 0.50 | 0.11 | asserted 0.93 |
| s3 | Following Beig and Schmidt we treat relativistic elasticity as a Lagrangian field theory. | method 0.99 | 0.79 | 0.02 | asserted 0.99 |
| s4 | The basic unknowns are mappings between spacetime and an abstract material manifold. | method 0.93 | 0.25 | 0.02 | asserted 1.00 |
| s5 | The equations of motion will be the Euler-Lagrange equations arising from the action principle. | method 0.98 | 0.38 | 0.03 | asserted 1.00 |
| s6 | Under certain (physically reasonable) restrictions on the elastic properties of the material the resulting system is hyperbolic and local existence results can be derived. | result 0.51 | 0.72 | 0.30 | asserted 0.86 |
| s7 | This way we can show well-posedness for the equations describing elastic matter on a fixed gravitational background, self-gravitating elastic matter and a finite elastic body on a fixed background. | result 0.91 | 0.92 | 0.64 | asserted 0.98 |
| s8 | The thesis also contains some results on energy conservation and on linearized elasticity. | contribution 0.80 | 0.89 | 0.05 | asserted 1.00 |
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The paper
Original abstract
There are many different formulations of relativistic elasticity. Most of them are only concerned with formal questions rather than questions regarding the PDE point of view. The aim of this thesis is to obtain various local existence results for dynamical scenarios involving elastic matter. Following Beig and Schmidt we treat relativistic elasticity as a Lagrangian field theory. The basic unknowns are mappings between spacetime and an abstract material manifold. The equations of motion will be the Euler-Lagrange equations arising from the action principle. Under certain (physically reasonable) restrictions on the elastic properties of the material the resulting system is hyperbolic and local existence results can be derived. This way we can show well-posedness for the equations describing elastic matter on a fixed gravitational background, self-gravitating elastic matter and a finite elastic body on a fixed background. The thesis also contains some results on energy conservation and on linearized elasticity.
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References
How the literature uses it
7 arXiv papers cite this in 14 sentences
Papers citing it · 8 sentences without a confident label
Cites it as background 5 papers
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We start from full relativistic elastodynamics, as laid down in [ref] , [ref] or [ref] .
Helical motion of elastic spheres · Relativistic Elastodynamics · 0.83
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For an extended discussion of the material representation in relativistic elasticity, see [ref] .
Dynamical compact elastic bodies in general relativity · Introduction · 0.87
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Since then, several authors have first aimed at proposing constitutive equations for Relativistic fluids [ref] and, then, at modeling Relativistic continuous media, most often at the astrophysics scale [ref] , for instance for the modeling of the solid crust of neutron stars, but also at a local scale [ref] , for mechanical engineering applications.
Souriau's Relativistic general covariant formulation of hyperelasticity revisited · Introduction · 1.00
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