On Hamiltonian continuum mechanics

Abstract

Continuum mechanics can be formulated in the Lagrangian frame (addressing motion of individual continuum particles) or in the Eulerian frame (addressing evolution of fields in an inertial frame). There is a canonical Hamiltonian structure in the Lagrangian frame. By transformation to the Eulerian frame we find the Poisson bracket for Eulerian continuum mechanics with deformation gradient (or the related distortion matrix). Both Lagrangian and Eulerian Hamiltonian structures are then discussed from the perspective of space-time variational formulation and by means of semidirect products and Lie algebras. Finally, we discuss the importance of the Jacobi identity in continuum mechanics and approaches to prove hyperbolicity of the evolution equations and their gauge invariance.

arXiv commentSubmitted to Physica D

References in corpus (22)

• High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids
• Structure and structure-preserving algorithms for plasma physics
• High order ADER schemes for continuum mechanics
• Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations
• On the Existence of a Maximal Cauchy Development for the Einstein Equations - a Dezornification
• Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity
• A new form of governing equations of fluids arising from Hamilton's principle
• Symmetric form for the hyperbolic-parabolic system of fourth-gradient fluid model
• Continuous and discrete Clebsch variational principles
• Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity
• Continuum mechanics with torsion
• A hierarchy of Poisson brackets in non-equilibrium thermodynamics
• Hamiltonian and Godunov Structures of the Grad Hierarchy
• A Numerical Scheme for Non-Newtonian Fluids and Plastic Solids under the GPR Model
• A new continuum model for general relativistic viscous heat-conducting media
• Solid-fluid dynamics of yield-stress fluids
• Hamiltonian dynamics on matched pairs
• Kinematic quantities of finite elastic and plastic deformation
• Extra mass flux in fluid mechanics
• Ehrenfest regularization of Hamiltonian systems
• On numerical methods for hyperbolic PDE with curl involutions
• A unified hyperbolic formulation for viscous fluids and elastoplastic solids

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