论文标题
球形和平面滚珠轴承 - 具有不变措施的非自然系统
Spherical and planar ball bearings -- nonholonomic systems with invariant measures
论文作者
论文摘要
我们首先构建了$ n $同质球$ \ mathbf b_1，\ dots，\ mathbf b_n $的非智能系统，带有中心$ o_1，...，o_n $，并且在固定sphey $ \ nater $ \ nath $ \ mathbf s_0 $ $ $ o $ o $ o $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $和RADIUS $ r $中。此外，假定一个动态非对称球$ \ mathbf s $ RADIUS $ r+2r $，并且中心与固定球的中心$ o $相吻合，而无需在移动球上滑过$ \ Mathbf b_1，\ dots，\ dots，\ dots，\ nathbf b_n $。我们证明这些系统具有不变的度量。作为第二个任务，我们考虑限制，当半径$ r $倾向于无限。我们获得了一个相应的平面问题，该问题由$ n $同质球组成$ \ mathbf b_1，\ dots，\ mathbf b_n $，带有中心$ o_1，...，o_n $和相同的半径$ r $，它们正在滚动而不会在固定飞机上滑倒$ pllape $ pllane $ plame $ notimen $σ$，而无需滑动，而不必滑动。我们证明，该系统具有不变的度量，并且根据Euler-Jacobi定理可以在四倍体中整合。
We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane $Σ_0$, and a moving plane $Σ$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler-Jacobi theorem.
