# On Hamiltonian continuum mechanics

## TL;DR

This paper explores the Hamiltonian formulation of continuum mechanics in both Lagrangian and Eulerian frames, deriving Poisson brackets and discussing mathematical structures and properties like the Jacobi identity.

## Contribution

It introduces a Hamiltonian structure in the Eulerian frame, extending the canonical Lagrangian Hamiltonian framework to Eulerian continuum mechanics.

## Key findings

- Derived the Poisson bracket for Eulerian continuum mechanics.
- Analyzed the Hamiltonian structures using space-time variational principles.
- Discussed the role of the Jacobi identity and hyperbolicity in continuum equations.

## 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.