Epicycles in hyperbolic sky

Olga Romaskevich

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arXiv · 1704.01339

Epicycles in hyperbolic sky

Olga Romaskevich

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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 ?

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Olga Romaskevich. 2017-04-05. Epicycles in hyperbolic sky. https://arxiv.org/abs/1704.01339

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