2019/07/31 by Michal Pavelka, Ilya Peshkov, Václav Klika +1 · 27 citations
Engineering · Mathematics · Physics and Astronomy · #Analytical mechanics #Classical mechanics #Continuum mechanics #Dynamics and Control of Mechanical Systems #Elasticity and Material Modeling #Eulerian path #Hamiltonian mechanics #Inertial frame of reference #Lagrangian #Lagrangian mechanics #Lie algebra #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Phase space #Physics #Poisson bracket #Pure mathematics #Quantum mechanics #Quantum statistical mechanics #math-ph #math.MP #physics.class-ph #physics.flu-dyn
paper · pdf · doi:10.1016/j.physd.2020.132510
published in Physica D Nonlinear Phenomena 408, 132510 (Elsevier BV) · Submitted to Physica D
arxiv created 2020/03/31 · openalex publication_date 2020/04/13 · arxiv updated 2020/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.