image/svg+xml Michal Pavelka, Ilya Peshkov, Václav Klika On Hamiltonian continuum mechanics Hyperbolicity Continuum Hamilton canonical equations Lagrangian frame Eulerian frame deformation tensor distortion Symmetric Hyperbolic Thermodynamically Compatible S. K. Godunov, E. Romenskii, I. Peshkov SHTC equations are Hamiltonian. de Gruyter, Berlin, 2018 Pavelka, Klika, Grmela Pavelka, Peshkov, Klika, On Hamiltonian continuum mechanics, arxiv 2019 General Hamiltonian evolution Poisson bivector Poisson bracket Leibniz rule, Jacobi identity, antisymmetric Transformation invariance Let be a conserved quantity, Then the evolution equations are invariant w.r.t. e.g. conservation of momentum -> w.r.t. translations angular momentum -> w.r.t. rotations Galilean booster -> Galilean transformations Dubrovin, Novikov → Riemannian structure metric tensor (in the state space) second covariant differential of energy Hamiltonian (assuming the metric be non-degenerate) evolution symmetric hyperbolicity Geometric mechanics Lie group → Euler-Poincaré equations Symmetric hyperbolic qusilinear equations Transformation invariance Proof of symmetric hyperbolicity Geometric generalizations (mixtures) Prague Toulouse Prague Particle mechanics General Equation for Non-Equilibrium Reversible-Irreversible Coupling GENERIC functional of state variables Poisson bracket energy dissipation potential entropy classical mechanics, kinetic theory, electromagnetism, fluid dynamics, elasticity, plasticity, viscoelasticity, mixtures, heat conduction, hyperbolic heat conduction, complex fluids, turbulence, ... reversible irreversible Miroslav GrmelaHans Christian Öttinger Peshkov, Pavelka, Romenski, Grmela, Hamiltonian and Godunov-type continuum mechanics and thermodynamics, Continuum mechanics and Thermodynamics 2018 M. Grmela, H. C. Öttinger., Rev. E 1997 Dzyalshinskii, Volovick, Annals of Physics 1980 Symmetric hyperbolicity? Godunov & Boillat for conservation laws Microscopic definition of phase volume? Relative velocity with self-advection? Geometric numerical schemes? dissipation
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