# Epicycles in hyperbolic sky

## TL;DR

This paper generalizes classical results about the asymptotic velocity of a swiveling arm to complete orientable Riemannian surfaces with non-zero curvature, including hyperbolic planes, extending celestial mechanics analogies.

## Contribution

It extends Lagrange's problem from Euclidean to arbitrary Riemannian surfaces, providing new insights into motion dynamics on curved geometries.

## Key findings

- Derived asymptotic velocity formulas for swiveling arms on curved surfaces
- Extended classical Euclidean results to hyperbolic and other non-zero curvature surfaces
- Connected geometric motion problems to celestial mechanics models

## 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 heliocentric epicycle model would make in one billion years ?