# Epicycles in the Hyperbolic Sky

- 2 Downloads

## 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?

## Keywords

Ergodic theory Lagrange problem Swivelling arms## Notes

### 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.

## References

- Bohl, P.: Über ein in der theorie der säkularen Störungen vorkommendes problem. J. Reine Angew. Math.
**135**, 189–283 (1909)MathSciNetzbMATHGoogle Scholar - Ghys, É.: Resonances and Small Divisors. Kolmogorov’s Heritage in Mathematics, pp. 187–213. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
- Hartman, P., Van Kampen, E.R., Wintner, A.: Mean motions and distribution functions. Am. J. Math.
**59**(2), 261–269 (1937)MathSciNetCrossRefzbMATHGoogle Scholar - Hausmann, J.-C.: Sur la Topologie des Bras Articulés, Algebraic Topology Poznań, Lecture Notes in Mathematics, pp. 146–159 (1989)Google Scholar
- Hausmann, J.-C.: Contrôle des bras articulés et transformations de Möbius. L’Enseignement Math.
**51**, 87–115 (2005)zbMATHGoogle Scholar - Jessen, B.: Some aspects of the theory of almost periodic functions. In: Proceedings of International Congress of Mathematicians, Amsterdam, 1, ICM, pp. 304–351 (1954)Google Scholar
- Jessen, B., Tornehave, H.: Mean motions and zeros of almost periodic functions. Acta Math.
**77**, 137–279 (1945)MathSciNetCrossRefzbMATHGoogle Scholar - Kapovich, M., Milson, J.: On the moduli space of polygons in the Euclidean plane. J. Differ. Geom.
**42**(1), 133–164 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - Kornfeld, I.P., Sinai, Ya G., Fomin, S.V.: Ergodic Theory. Springer, Berlin (1982)CrossRefGoogle Scholar
- Lagrange, J.-L.: Théorie des Variations séculaires des éléments des Planètes, I, II, Nouveaux Mémoires de l’Académie de Berlin, 5 (1781, 1782)Google Scholar
- Poincaré, H.: La Valeur de la Science, Flammarion, Paris (1911)Google Scholar
- Sakai, T.: Riemannian Geometry. American Mathematical Society, New York (1996)CrossRefzbMATHGoogle Scholar
- Weyl, H.: Mean motion. Am. J. Math.
**60**, 889–896 (1938)MathSciNetCrossRefzbMATHGoogle Scholar - Zvonkine, D.: Configuration spaces of hinge constructions. Russ. J. Math. Phys.
**5**, 2 (1997)MathSciNetzbMATHGoogle Scholar