arXiv 1302.06742013-02math-phmath.MP

仿射运动力学中的约束和对称性

Constraints and symmetry in mechanics of affine motion

Jan Jerzy Sławianowski, Barbara Gołubowska, Vasyl Kovalchuk

备注：: 33 pages
发表信息：: Journal of Geometry and Physics 78 (2014), 59-79
DOI：: 10.1016/j.geomphys.2014.01.012

摘要：

本文的目的是对相邻刚体动力学中出现的问题进行更深入的几何分析。首先给出了仿射运动的极分解和双极分解的几何解释。稍后将回顾对仿射运动施加的一些附加约束，包括完整约束和非完整约束。特别是，我们集中在某些自然的非完整模型的旋转少的运动。我们讨论了通常的d'Alembert模型和vakonomic动力学。由此产生的方程是完全不同的。目前还不清楚哪种模式实际上更好。在任何情况下，它们都不同于完整约束，完整约束将无旋转运动定义为与时间相关的对称放置矩阵族。后一种模型似乎是非几何和非物理的。然而，我们的非完整模型和极分解之间有一定的关系。

英文摘要：

The aim of this paper is to perform a deeper geometric analysis of problems appearing in dynamics of affinely rigid bodies. First of all we present a geometric interpretation of the polar and two-polar decomposition of affine motion. Later on some additional constraints imposed on the affine motion are reviewed, both holonomic and non-holonomic. In particular, we concentrate on certain natural non-holonomic models of the rotation-less motion. We discuss both the usual d'Alembert model and the vakonomic dynamics. The resulting equations are quite different. It is not yet clear which model is practically better. In any case they both are different from the holonomic constraints defining the rotation-less motion as a time-dependent family of symmetric matrices of placements. The latter model seems to be non-geometric and non-physical. Nevertheless, there are certain relationships between our non-holonomic models and the polar decomposition.

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