仿射体系统力学。结构介质动力学中的几何基础及其应用
Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media
摘要：
在本文中，我们研究仿射刚体系统的力学，即，在仿射几何意义上的刚体。在分子晶体、颗粒介质和其他物理对象的建模中，某些物理应用是可能的。特别有趣的是动力学模型不变下的组基本几何的自由度。与单体情况相反，存在非平凡的势在该组下不变（左作用和右作用）。讨论了仿射体对的相对（互）变形张量的概念。标量不变量构建这样的张量。与单个仿射体的变形标量相比，存在本质新颖性，即，存在相互变形的仿射不变标量。因此，层次的相互作用模型，根据其不变性组，从欧几里德仿射的，可以考虑。
英文摘要：
In the present paper we investigate the mechanics of systems of affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry. Certain physical applications are possible in modelling of molecular crystals, granular media, and other physical objects. Particularly interesting are dynamical models invariant under the group underlying geometry of degrees of freedom. In contrary to the single body case there exist nontrivial potentials invariant under this group (left and right acting). The concept of relative (mutual) deformation tensors of pairs of affine bodies is discussed. Scalar invariants built of such tensors are constructed. There is an essential novelty in comparison to deformation scalars of single affine bodies, i.e., there exist affinely-invariant scalars of mutual deformations. Hence, the hierarchy of interaction models according to their invariance group, from Euclidean to affine ones, can be considered.
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