论哈密顿连续介质力学
On Hamiltonian continuum mechanics
摘要：
连续统力学可以在拉格朗日框架（解决单个连续统粒子的运动）或欧拉框架（解决惯性框架中的场的演化）中制定。在拉格朗日标架中存在一个正则哈密顿结构。通过变换到欧拉标架，我们找到了具有变形梯度的欧拉连续介质力学的泊松括号（或相关的畸变矩阵）。拉格朗日和欧拉的哈密顿结构，然后讨论从时空变分的角度，并通过半直积和李代数。最后，讨论了Jacobi恒等式在连续介质力学中的重要性以及证明演化方程双曲性和规范不变性的方法。
英文摘要：
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.
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