## Abstract

Consider a swiveling arm on an oriented complete riemannian surface composed of three geodesic intervals, attached one to another in a chain. Each interval of the arm rotates with constant angular velocity around its extremity contributing to a common motion of the arm. Does the extremity of such a chain have an asymptotic velocity? This question for the motion in the euclidian plane, formulated by J.-L. Lagrange, was solved by P. Hartman, E. R. Van Kampen, A. Wintner. We generalize their result to motions on any complete orientable surface of non-zero (and even non-constant) curvature. In particular, we give the answer to Lagrange’s question for the movement of a swiveling arm on the hyperbolic plane. The question we study here can be seen as a dream about celestial mechanics on any riemannian surface: how many turns around the Sun a satellite of a planet in the geliocentric epicycle model would make in 1 billion years?

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## Notes

- 1.
*puisque nous sommes en train de faire des hypothèses, une hypothèse de plus ne nous coûtera pas davantage.* - 2.
*L’astronomie est utile, parce qu’elle nous élève au-dessus de nous-mêmes ; elle est utile, parce qu’elle est grande; elle est utile parce qu’elle est belle; voilà ce qu’il faut dire.* - 3.
“

*It is hard and maybe even impossible to say something on the nature of angle*\(\varphi \)*in the general case*” (English translation). Lagrange’s angle \(\varphi \) is the continuous branch of the argument \(\varphi (t)\) defined above. - 4.
The same assumptions about \(\omega _j\) hold for the Theorem 3 and the swiveling arm on the hyperbolic plane.

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## Acknowledgements

I am grateful to Anatoly Stepin for sharing with me the question about the asymptotic angular velocity of a swiveling arm on the hyperbolic plane when I was a student at Moscow State University. I thank Étienne Ghys for very fruitful discussions that helped me change the approach of this question and drastically simplify the arguments. I also thank Bruno Sevennec as well as the anonymous referee for pertinent questions and remarks that helped me improve the text. The principal part of this work was accomplished when I was a graduate student at the UMPA laboratory at École Normale Supérieure de Lyon. I thank my reporters, François Beguin and Alain Chenciner, for their comments. During the period of the work on this project, I was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR) as well as by a personal grant l’Oréal-UNESCO for Women in Science 2016.

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In memory of my grandfather V. V. Beletskii, a mathematician and a poet.

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Paris-Romaskevich, O. Epicycles in the Hyperbolic Sky.
*Arnold Math J.* **4, **251–277 (2018). https://doi.org/10.1007/s40598-019-00107-w

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### Keywords

- Ergodic theory
- Lagrange problem
- Swivelling arms