论文标题
球形和平面球轴承 - 对可整合病例的研究
Spherical and Planar Ball Bearings -- a Study of Integrable Cases
论文作者
论文摘要
我们考虑$ n $同质球$ \ mathbf b_1，\ dots，\ mathbf b_n $的非单位系统系统，其半径$ r $在没有固定的球体$ \ MATHBF S_0 $的情况下滚动，带有中心$ O $和RADIUS $ R $。此外，假设一个动态非对称球$ \ mathbf s $与中心与固定球的中心$ o $相吻合，而$ \ mathbf s_0 $ rolls rolls却不在与移动球$ \ mathbf b_1，\ mathbf b_n $接触的情况下接触。该问题以四种不同的配置考虑。我们得出运动方程，并证明这些系统具有不变的度量。作为主要结果，对于$ n = 1 $，我们发现了两个根据Euler-Jacobi定理在四倍体中集成的情况。可获得的可集成的非自我工艺模型是众所周知的Chaplygin Ball-nevely问题的自然扩展。此外，我们明确地集成了平面问题，这些问题由相同半径的$ n $均质球组成，但具有不同的质量，这些球不在固定平面上滑倒$σ_0$的滚动，而平面$σ$在没有滑过这些球上移动的平面$σ$。
We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping in contact to the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations. We derive the equations of motion and prove that these systems possess an invariant measure. As the main result, for $n=1$ we found two cases that are integrable in quadratures according to the Euler-Jacobi theorem. The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems. Further, we explicitly integrate the planar problem consisting of $n$ homogeneous balls of the same radius, but with different masses, that roll without slipping over a fixed plane $Σ_0$ with a plane $Σ$ that moves without slipping over these balls.
