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Gyroscopic Chaplygin Systems and Integrable Magnetic Flows on Spheres

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Abstract

We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slip** and twisting over a plane and over a sphere in \({\mathbb {R}}^n\) as examples of gyroscopic SO(n)-Chaplygin systems. We describe an invariant measure and provide examples of \(SO(n-2)\)-symmetric systems (ball with gyroscope) that allow the Chaplygin Hamiltonization. In the case of additional SO(2)-symmetry, we prove that the obtained magnetic geodesic flows on the sphere \(S^{n-1}\) are integrable. In particular, we introduce the generalized Demchenko case in \({\mathbb {R}}^n\), where the inertia operator of the system is proportional to the identity operator. The reduced systems are automatically Hamiltonian and represent the magnetic geodesic flows on the spheres \(S^{n-1}\) endowed with the round-sphere metric, under the influence of a homogeneous magnetic field. The magnetic geodesic flow problem on the two-dimensional sphere is well known, but for \(n>3\) was not studied before. We perform explicit integrations in elliptic functions of the systems for \(n=3\) and \(n=4\) and provide the case study of the solutions in both situations.

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Notes

  1. Demchenko’s PhD advisor, Anton Bilimović (1879–1970), was a distinguished student of Peter Vasilievich Voronec (1871–1923) and one of the founders of Belgrade’s Mathematical Institute. We note that some recent results [see Borisov and Tsiganov (2020); Borisov et al. (2021)] are inspired by Bilimović’s work in nonholonomic mechanics (Bilimovitch 1913a, b, 1914; Bilimovic 1915; Bilimovich 1916).

  2. One can compare the form of Eq. (2.4) with the compact form of the Voronec equations obtained from the Voronec principle, see, e.g., Dragović et al. (2023).

  3. Let us note that in Ehlers et al. (2005), the term “JK" is used for the associated semi-basic two-form \(\sigma \) on \(T^*S\) given below.

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Acknowledgements

We are very grateful to the referees for valuable remarks that significantly helped us to improve the exposition, and in particular for providing us an example which led to Remark 5.2. We thank Viswanath Ramakrishna for reading the manuscript and providing a feedback. This research has been supported by the Project No. 7744592 MEGIC “Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics” of the Science Fund of Serbia, Mathematical Institute of the Serbian Academy of Sciences and Arts and the Ministry for Education, Science, and Technological Development of Serbia, and the Simons Foundation Grant No. 854861.

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  1. Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, USA

    Vladimir Dragović

  2. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Vladimir Dragović, Borislav Gajić & Božidar Jovanović

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  1. Vladimir Dragović
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Dragović, V., Gajić, B. & Jovanović, B. Gyroscopic Chaplygin Systems and Integrable Magnetic Flows on Spheres. J Nonlinear Sci 33, 43 (2023). https://doi.org/10.1007/s00332-023-09901-5

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