  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.
