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Regul. Chaotic Dyn., 2019, том 24, выпуск 5, страницы 511–524 (Mi rcd1024)  

Эта публикация цитируется в 1 научной статье (всего в 1 статье)

Sergey Chaplygin Memorial Issue

Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball

Miguel D. Bustamante, Peter Lynch

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

Аннотация: 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.

Ключевые слова: Noether’s theorem, nonholonomic systems, symmetry, Routh sphere, Chaplygin ball


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Тип публикации: Статья
MSC: 37J60, 70E18, 70F25, 70H07, 70H33
Поступила в редакцию: 31.05.2019
Принята в печать:27.08.2019
Язык публикации: английский

Образец цитирования: Miguel D. Bustamante, Peter Lynch, “Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball”, Regul. Chaotic Dyn., 24:5 (2019), 511–524

Цитирование в формате AMSBIB
\by Miguel D. Bustamante, Peter Lynch
\paper Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 5
\pages 511--524

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    Эта публикация цитируется в следующих статьяx:
    1. A. V. Tsiganov, “On a time-dependent nonholonomic oscillator”, Russ. J. Math. Phys., 27:3 (2020), 399–409  crossref  mathscinet  zmath  isi  scopus
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