The dynamics of a spherical body with a non-uniform mass distribution rolling on a plane were discussed by Sergey Chaplygin, whose 150th birthday we celebrate this year. The Chaplygin top is a non-integrable system, with a colourful range of interesting motions. A special case of this system was studied by Edward Routh, who showed that it is integrable. The Routh sphere has a centre of mass offset from the geometric centre, but it has an axis of symmetry through both these points, and equal moments of inertia about all axes orthogonal to the symmetry axis. There are three constants of motion: the total energy and two quantities involving the angular momenta.It is straightforward to demonstrate that these quantities, known as the Jellett and Routh constants, are integrals of the motion. However, their physical significance has not been fully understood. In this paper, we show how the integrals of the Routh sphere arise from Emmy Noether’s invariance identity. We derive expressions for the infinitesimal symmetry transformations associated with these constants. We find the finite version of these symmetries and provide their geometrical interpretation.As a further demonstration of the power and utility of this method, we find the Noetherian symmetries and corresponding integrals for a system introduced recently, the Chaplygin ball on a rotating turntable, confirming that the known integrals are directly obtained from Noether’s theorem.

中文翻译：

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Routh球和Chaplygin球的非完整Noether对称性和积分

谢尔盖·查普林金（Sergey Chaplygin）讨论了球体动力学的问题，该球状体在飞机上的质量分布不均匀，我们今年庆祝了他的150岁生日。Chaplygin顶部是一个不可集成的系统，具有一系列有趣的动作。爱德华·罗斯（Edward Routh）研究了该系统的一个特例，他证明了该系统是可集成的。Routh球的质量中心偏离几何中心，但是它具有通过这两个点的对称轴，并且绕正交于对称轴的所有轴具有相等的惯性矩。运动有三个常数：总能量和两个涉及角动量的量，可以很容易地证明这些量（称为Jellett和Routh常数）是运动的积分。然而，它们的物理意义尚未完全了解。在本文中，我们展示了Routh球面的积分是如何由Emmy Noether的不变恒等式产生的。我们导出与这些常数关联的无穷小对称变换的表达式。我们找到了这些对称性的有限形式，并提供了它们的几何解释。为进一步证明此方法的功效和效用，我们找到了最近引入的系统（旋转转盘上的Chaplygin球）的Noetherian对称性和相应的积分，证实了已知积分是直接从Noether定理获得的。我们导出与这些常数关联的无穷小对称变换的表达式。我们找到了这些对称性的有限形式，并提供了它们的几何解释。为进一步证明此方法的功效和效用，我们找到了最近引入的系统（旋转转盘上的Chaplygin球）的Noetherian对称性和相应的积分，证实了已知积分是直接从Noether定理获得的。我们导出与这些常数关联的无穷小对称变换的表达式。我们找到了这些对称性的有限形式，并提供了它们的几何解释。为进一步证明此方法的功效和效用，我们找到了最近引入的系统（旋转转盘上的Chaplygin球）的Noetherian对称性和相应的积分，证实了已知积分是直接从Noether定理获得的。